I agree with DeweyCA's comments that Christian's are not to be vengeful. At the same time Christians are to seek and support justice.
In the Sermon on the Mount the context speaks of a debt or obligation. Even the If you are compelled to walk a mile, give another mile teaching has been described as set in the context of the Roman soldiers legal right to compel passers by to carry his pack a mile.
The substance of the teaching is that when you have an obligation/debt then don't just pay off the obligation or debt but go the extra mile, offer the other cheek. It goes beyond obligation to reconciliation.
While this does address "private" relationships, it does not include all private relationships. It does not address the right to self-defense. I don't think that it tells us that we are to turn the other cheek when a member of family is attacked by another in our presence.
Their is much to understand here, but a universal application of "Turn the other cheek" like "Judge not that you may not be judged" is to contradict the larger theme of "Thy Kingdom come" and the manifestation of Gods glory and character on the earth.
A dispassionate effort is worthwhile, of course.