This verse is easy to misunderstand without an understanding of the culture that produced it. It helps to read it very literally: If someone strikes you on the right cheek, turn to him the other also.
Why is that significant? Think back to the Old Testament, where it would explicitly point out anytime someone was left-handed, because that was thought to be really weird. Being right-handed was the norm, and that's an important part of this verse.
It would be very awkward for a right-handed person bent on doing harm to attack someone standing in front of them by striking them on the right side of their face. The only convenient strike that works this way is a backhanded slap, which is very different in nature: an insult, not an assault. Understanding this changes the meaning of the verse greatly:
Do not let evil men provoke you. If someone insults you with a backhanded slap, do not respond to the provocation, but turn your other cheek instead.
This is in the same character as the other verses around it, particularly the one about being forced to go a mile with someone (which a Roman citizen could legally force a Jew to do). There were Jews at that time who thought their religious duty was to resist Roman occupation in any way they could, including by killing Romans. If the followers of Christ got mixed up in that, they would have ended up sharing his fate, execution at Roman hands, which would have seriously impeded the church's ability to grow. So the message is, make it absolutely clear that you're going along with the law, so no one has any excuse to try to use the law to make even more trouble for you.
Also, keep in mind that when they were actually in real danger from people who were determined not to let the law get in the way of what they wanted:
Luke 22: 35-36
35 he said unto them, When I sent you without purse, and scrip, and shoes, lacked ye any thing? And they said, Nothing.
36 Then said he unto them, But now, he that hath a purse, let him take it, and likewise his scrip: and he that hath no sword, let him sell his garment, and buy one.
Two different directives for two different points in time, but both with the same objective: to keep his disciples safe from harm. These two principles should provide a firm foundation for a Christian to base his understanding of self-defense on.