Questions about the Sermon on the Mount are always challenging. Here is one I just received: 

How would you exegete this passage (Matt 5:38-40)? I have read  that in Jewish culture, a slap to the face is considered an insult. Jesus isn’t saying let your face be a punching bag, but if someone insults you, let it go. Is this correct? Love Grace in Focus radio.                

It is a tough text with little contextual help. 

Most likely the context concerns suffering during persecution. 

The big idea here is that we are not to retaliate over a wrong a person does.

Remember when during His trials, Jesus was struck with the palm of the hand (Matt 26:67-68; 27:30, and parallels)? The text does not say that He turned the other cheek. If Jesus’ point about turning the other cheek were meant literally, then we know He did turn the other cheek. But it is more likely that He was speaking figuratively. See John 18:23 (and Acts 23:3 concerning Paul when he was struck).

I think the point is that when we are persecuted, we are not to fight back. Jesus did not fight back when they struck Him, spit on Him, mocked Him, etc. Peter reminds us that “When He was reviled, He did not revile in return; when He suffered, He did not threaten, but committed Himself to Him who judges righteously…” (1 Pet 2:23).

The point does not concern self-defense. Obviously, we are to defend ourselves and our family if attacked (Matt 24:43). 

The point does not concern the military or the police, either.  

Verse 40 (“If anyone wants to sue you and take away your tunic, let him have your cloak also”) is not talking about robbery. Instead, it likely refers to someone who is using legal means to persecute you. In that case, we are not to resist.

Some see these verses as an expression of a kingdom ethic, that is, this is the way things will be during the Millennium. But there will not be much if any persecution of believers then. More likely this concerns first century persecution and persecution during the Tribulation. But, in addition, it is a general instruction which applies to persecution in any age. We are not to retaliate when people persecute us for our faith.

Here are some blogs on this issue (one by TGC, one by Gotquestions.org, and one by Focus on the Family) that you might find helpful.