So what does a slap in the face mean? A slap isn't always a literal slap. It should be known that a slap may also mean an insult or name-calling. The Jewish culture in Jesus' day had already misused and abused the eye for eye and tooth for tooth rule in Exodus 21:24. Instead of using it as a standard for punishing offenses, they have misused it in order to pursue personal revenge or think that God allows personal revenge. They took the whole verse out of context ignoring verses in the Old Testament that clearly forbade revenge. Did they even read Deuteronomy 32:39 that says that revenge is not ours but God's? It's probable that their culture that day that even a mere insult already warranted someone the right to hit the one who threw the insult.
So what did this all mean to turn the other cheek? This was really all about not getting into a fight when somebody calls you a name. It's like if somebody insults you then let it go. It wasn't life threatening. Why get into a fistfight because somebody called you a name? True, it hurts but why waste time having lawsuits over it? Just imagine how many bigger cases will be left unattended if everyone kept having lawsuits over trivial cases. Violence is only permissible in the act of self-defense but never in the act because somebody insulted you. Is an insult life threatening? No. But you certainly have the right to lawfully sue someone for trying to accuse you of a crime that could have falsely sentenced you to death (Deuteronomy 19:18-19).
It would also be good to discuss verses 40 to 41 to further clarify things. What does it mean to sue someone for your coat? Commentator John Gill would describe it as not to get into a violent brawl over your personal property. This doesn't mean you don't have the right to report to someone you've been ripped or warn others. But this means that sometimes, it's better to lose something than to get into a fight over something. As for verse 41, we can further explain it with the setting. Roman soldiers could ask anyone to carry their gear for a mile. By being willing to carry their gear another mile would mean defeating the Roman soldiers' roughness with kindness. But this doesn't mean surrendering one's rights to abusive officials while discouraging one from taking the law into one's hands.