I made progress more gradually--went from 300 to 1200 baud, and felt like I was getting real fast data! (On my commodore 64, no less.)
That blazing 300 baud was accomplished by using an "acoustic coupler." The modem had two rubber cups on it. You stuck the phone handset's receiver and transmitter in the cups.
And if you sneezed or there was a loud noise the modem would "hear" it and spew gibberish all over your screen.
Oh, and ALL phone handsets were the same so they all fit the rubber cups. Then Ma Bell came out with the Princess phone and it wouldn't work :)