Perhaps we should quote the words of those closest to the period (this list by no means purports to be exhaustive. Feel free to cite other early writers on the subject).
It is within the power of all, therefore, in every Church, who may wish to see the truth, to contemplate clearly the tradition of the apostles manifested throughout the whole world; and we are in a position to reckon up those who were by the apostles instituted bishops in the Churches, and [to demonstrate] the succession of these men to our own times; those who neither taught nor knew of anything like what these [heretics] rave about. ... Since, however, it would be very tedious, in such a volume as this, to reckon up the successions of all the Churches, we do put to confusion all those who, in whatever manner, whether by an evil self-pleasing, by vainglory, or by blindness and perverse opinion, assemble in unauthorized meetings; [we do this, I say,] by indicating that tradition derived from the apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul; as also [by pointing out] the faith preached to men, which comes down to our time by means of the successions of the bishops. For it is a matter of necessity that every Church should agree with this Church, on account of its preeminent authority -- that is, the faithful everywhere -- inasmuch as the Apostolic Tradition has been preserved continuously by those who are everywhere. ... In this order, and by this succession, the ecclesiastical tradition from the apostles, and the preaching of the truth, have come down to us. And this is most abundant proof that there is one and the same vivifying faith, which has been preserved in the Church from the apostles until now, and handed down in truth. ...Since therefore we have such proofs, it is not necessary to seek the truth among others which it is easy to obtain from the Church; since the apostles, like a rich man [depositing his money] in a bank, lodged in her hands most copiously all things pertaining to the truth: so that every man, whosoever will, can draw from her the water of life. For she is the entrance to life; all others are thieves and robbers. On this account are we bound to avoid them, but to make choice of the thing pertaining to the Church with the utmost diligence, and to lay hold of the tradition of the truth. For how stands the case? Suppose there arise a dispute relative to some important question among us, should we not have recourse to the most ancient Churches with which the apostles held constant intercourse, and learn from them what is certain and clear in regard to the present question? For how should it be if the apostles themselves had not left us writings? Would it not be necessary, [in that case,] to follow the course of the tradition which they handed down to those to whom they did commit the Churches?
Since, therefore, the tradition from the apostles does thus exist in the Church, and is permanent among us, let us revert to the Scriptural proof furnished by those apostles who did also write the Gospel, in which they recorded the doctrine regarding God, pointing out that our Lord Jesus Christ is the truth, and that no lie is in Him. (St. Irenaeus of Lyons, Adversus Haereses, III, 3-5)
Our appeal, therefore, must not be made to the Scriptures; nor must controversy be admitted on points in which victory will either be impossible, or uncertain, or not certain enough. But even if a discussion from the Scriptures should not turn out in such a way as to place both sides on a par, (yet) the natural order of things would require that this point should be first proposed, which is now the only one which we must discuss: "With whom lies that very faith to which the Scriptures belong. From what and through whom, and when, and to whom, has been handed down that rule, by which men become Christians?" For wherever it shall be manifest that the true Christian rule and faith shall be, there will likewise be the true Scriptures and expositions thereof, and all the Christian traditions. ... If, then, these things are so, it is in the same degree manifest that all doctrine which agrees with the apostolic churches—those moulds and original sources of the faith must be reckoned for truth, as undoubtedly containing that which the (said) churches received from the apostles, the apostles from Christ, Christ from God. Whereas all doctrine must be prejudged as false which savours of contrariety to the truth of the churches and apostles of Christ and God. (Tertullian, On the Prescription of Heretics, XIX, XXI)
On the whole, then, if that is evidently more true which is earlier, if that is earlier which is from the very beginning, if that is from the beginning which has the apostles for its authors, then it will certainly be quite as evident, that that comes down from the apostles, which has been kept as a sacred deposit in the churches of the apostles. Let us see what milk the Corinthians drank from Paul; to what rule of faith the Galatians were brought for correction; what the Philippians, the Thessalonians, the Ephesians read by it; what utterance also the Romans give, so very near (to the apostles), to whom Peter and Paul conjointly bequeathed the gospel even sealed with their own blood. We have also St. John's foster churches. For although Marcion rejects his Apocalypse, the order of the bishops (thereof), when traced up to their origin, will yet rest on John as their author. In the same manner is recognised the excellent source of the other churches. I say, therefore, that in them (and not simply such of them as were rounded by apostles, but in all those which are united with them in the fellowship of the mystery of the gospel of Christ ) that Gospel of Luke which we are defending with all our might has stood its ground from its very first publication; whereas Marcion's Gospel is not known to most people, and to none whatever is it known without being at the same time condemned. It too, of course, has its churches, but specially its own----as late as they are spurious; and should you want to know their original, you will more easily discover apostasy in it than apostolicity, with Marcion forsooth as their founder, or some one of Marcion's swarm. Even wasps make combs; so also these Marcionites make churches. The same authority of the apostolic churches will afford evidence to the other Gospels also, which we possess equally through their means, and according to their usage----I mean the Gospels of John and Matthew----whilst that which Mark published may be affirmed to be Peter's whose interpreter Mark was. For even Luke's form of the Gospel men usually ascribe to Paul. And it may well seem that the works which disciples publish belong to their masters. Well, then, Marcion ought to be called to a strict account concerning these (other Gospels) also, for having omitted them, and insisted in preference on Luke; as if they, too, had not had free course in the churches, as well as Luke's Gospel, from the beginning. Nay, it is even more credible that they existed from the very beginning; for, being the work of apostles, they were prior, and coeval in origin with the churches themselves. But how comes it to pass, if the apostles published nothing, that their disciples were more forward in such a work; for they could not have been disciples, without any instruction from their masters? If, then, it be evident that these (Gospels) also were current in the churches, why did not Marcion touch them----either to amend them if they were adulterated, or to acknowledge them if they were uncorrupt? For it is but natural that they who were perverting the gospel, should be more solicitous about the perversion of those things whose authority they knew to be more generally received. Even the false apostles (were so called) on this very account, because they imitated the apostles by means of their falsification. In as far, then, as he might have amended what there was to amend, if found corrupt, in so far did he firmly imply that all was free from corruption which he did not think required amendment. In short, he simply amended what he thought was corrupt; though, indeed, not even this justly, because it was not really corrupt. For if the (Gospels) of the apostles have come down to us in their integrity, whilst Luke's, which is received amongst us, so far accords with their rule as to be on a par with them in permanency of reception in the churches, it clearly follows that Luke's Gospel also has come down to us in like integrity until the sacrilegious treatment of Marcion. In short, when Marcion laid hands on it, it then became diverse and hostile to the Gospels of the apostles. I will therefore advise his followers, that they either change these Gospels, however late to do so, into a conformity with their own, whereby they may seem to be in agreement with the apostolic writings (for they are daily retouching their work, as daily they are convicted by us); or else that they blush for their master, who stands self-condemned either way----when once he hands on the truth of the gospel conscience smitten, or again subverts it by shameless tampering. Such are the summary arguments which we use, when we take up arms against heretics for the faith of the gospel, maintaining both that order of periods, which rules that a late date is the mark of forgers, and that authority of churches which lends support to the tradition of the apostles; because truth must needs precede the forgery, and proceed straight from those by whom it has been handed on. (Tertullian, Against Marcion, IV, 5:1-7)
Since, in the beginning of this work, we promised to give, when needful, the words of the ancient presbyters and writers of the Church, in which they have declared those traditions which came down to them concerning the canonical books, and since Irenaeus was one of them, we will now give his words and, first, what he says of the sacred Gospels ... (Eusebius of Caesarea, Church History, V, 8)
And since these things are so, the judgment of the book of Wisdom ought not to be repudiated, since for so long a course of years that book has deserved to be read in the Church of Christ from the station of the readers of the Church of Christ, and to be heard by all Christians, from bishops downwards, even to the lowest lay believers, penitents, and catechumens, with the veneration paid to divine authority. (St. Augustine, On the Perseverance of the Saints, §27)
But let us now go back to consider the third step here mentioned, for it is about it that I have set myself to speak and reason as the Lord shall grant me wisdom. The most skillful interpreter of the sacred writings, then, will be he who in the first place has read them all and retained them in his knowledge, if not yet with full understanding, still with such knowledge as reading gives,--those of them, at least, that are called canonical. For he will read the others with greater safety when built up in the belief of the truth, so that they will not take first possession of a weak mind, nor, cheating it with dangerous falsehoods and delusions, fill it with prejudices adverse to a sound understanding. Now, in regard to the canonical Scriptures, he must follow the judgment of the greater number of catholic churches; and among these, of course, a high place must be given to such as have been thought worthy to be the seat of an apostle and to receive epistles.Accordingly, among the canonical Scriptures he will judge according to the following standard: to prefer those that are received by all the catholic churches to those which some do not receive. Among those, again, which are not received by all, he will prefer such as have the sanction of the greater number and those of greater authority, to such as are held by the smaller number and those of less authority. If, however, he shall find that some books are held by the greater number of churches, and others by the churches of greater authority (though this is not a very likely thing to happen), I think that in such a case the authority on the two sides is to be looked upon as equal. (St. Augustine, On Christian Doctrine, II, 8:12)
They arrived at it based on the spiritual nature of it's content, it's proximity to an Apostolic figure, and it's universal acceptance.
The Letter to the Hebrews, the Revelation of John, or 2 and 3 John had "universal acceptance" among orthodox Christians? Do tell.
PS: What else is "universal acceptance" but an appeal to the authority of the Church? I'm surprised you don't see this.
"PS: What else is "universal acceptance" but an appeal to the authority of the Church? I'm surprised you don't see this"
... but not an appeal to the heirarchy - i.e. the "authoritative, infallable Church". Rather an appeal to universal acceptance among all Christians.