Summary
YES, Christians have the right to defend themselves. When understood correctly, the verses cited don't negate this right but to show examples of the right mindset when defending ourselves, i.e. not with equal / greater violence in retaliation, but with grace and love (when situations warrant them) so the "enemies" can see the light of the Kingdom through our interaction with them.
These verses should not be misapplied into ruling out all wars or violence. Jesus was not a Pacifist. His allowing himself to be abused, tortured, and shamed on the cross was not a teaching so Christians shouldn't defend themselves, but to fulfill certain prophecies like Isa 53. At best, Jesus applied these verses by showing us how we are the enemies (of God) and thus "stop us in our tracks" of our life of violence and rebellion to be stunned and touched deeply at how God Himself loves us (the enemies) by substituting our sin with His grace.
Danger of overgeneralization
Merely citing Bible verses, especially when citing them out of context and generalizing the application to a whole range of human situations is not responsible Bible interpretation. For example: how does any particular Bible verse directly applicable to someone (who is clearly a non-combatant civilian) defending his family from a Russian soldier barging into his home in Ukraine or to someone using a gun to protect his family when he is massively outnumbered in an armed home robbery in which the gang of assailants have shown clear intent to kill him and the police is 30 minute away?
Situations matter!! The points below should highlight several aspects to be considered when discerning whether the use of violence is justified in a particular situation:
There is a time to acquiesce to be martyred for our faith, but that is not the right application of these verses, but of Matt 10:28,33. In the soldier and robber examples above, neither asks us to renounce our faith.
There is a time for "ministering in the area of reconciliation, peacemaking and restorative justice" (DJClayworth's answer) which I agree that it's exactly how these verses should be applied. The question is: when your life is in imminent danger (such as in the 2 examples above), when all other options except violence are no longer "live options", is this a time for peacemaking? If, on the other hand, you manage to disarm the assailant and bind him securely (using some non-lethal violence), this can be the time to minister, showing grace by not retaliating like a mafia.
We shouldn't abuse this right into justifying horrific retaliation, as I try to show below how the verses (properly interpreted) are meant to teach that retaliatory violence in itself should not be the first option to take in a personal affront. Example: when the enemy shows a clear intent to kill us, Christians have the right to use force for self-defense as a last resort.
It also does not mean we are obligated to take the path of violence, even when it is justified. It is simply an option that a Christian can legitimately take for self defense, or for protecting your loved ones, without violating those verses. For example, let's say the Holy Spirit speaks strongly to you (the oppressed) that the armed robber (whom God knows in advance that he is one of the elect) will spare your life, and that this is an opportunity for you to present the gospel. But when this Holy Spirit assurance is not given, you are justified to use violence as a last resort.
Proper context: Sermon on the Mount, not pacifism
The 2 verses you cited are from Jesus's Sermon on the Mount (Matthew's version) or Sermon on the Plain (Luke's version), and therefore the larger context (Matt 5-7) is about the higher law of grace and love expected from people belonging to the new covenant kingdom. The more immediate context (Matt 5:38-48) is about personal ethics in dealing with difficult people who are not gracious, loving, and just to us (i.e. our "enemies"). Kingdom people need to be salt and light to show God's way of grace and love to non-Kingdom people. Thus, the proper interpretive framework is how to differentiate us from them through our response to an affront.
It is not to advocate pacifism, i.e. refusing to fight in a just war or for protecting our territory. It is also not to command us to be a doormat, nor to prevent us from demanding or pursuing our rights to be treated humanly. Thus, the 2 verses you cited need to be interpreted in the context of how to behave differently than what people usually do: revenge (equal or even more severely) or love only their friends.
Did Jesus himself ever give an example that if someone is about to kill you, you should just let him? "Do not resist an evil person" does NOT mean giving someone a blank check to rape you, torture you, or murder you. Notice that none of the 4 examples in vv. 39-42 are a matter of life and death !! At the worst case, your cheek is sore for a day, you are naked until you have money to buy another tunic, you are tired of walking the other mile, and you don't get your money back from loaning a person. In none of the 4 cases is your life in danger! You get to live to see your enemy potentially become Christian as a result of showing grace to them.
The above context hopefully makes this command to make sense, and the cultural context below will make the command even more rational as well as gives us a much better idea on how to apply it to 21st century cultures.
Cultural context of the two verses
Much has been written on the 3 examples mentioned in Matt 5:39-41 about offering the left cheek, giving the coat also, and walking the 2nd mile. Should we be surprised that the original audience understood the 3 examples very differently than us, and as 21st century Bible interpreter we should first try to find the original cultural meaning before applying them to our 21st century culture?
From a 2016 Unsystematic theology blog article Turn the Other Cheek? (Explained in Context) quoting from a well regarded Protestant textbook on Kingdom Ethics (by Glen Stassen and David Gushee):
Doesn't Jesus' teaching encourage docile passivity, easy acquiescence to injustice and violence?
The answer? No.
Understood in it’s context, this teaching should be seen as an expression of nonviolent activism. Glen Stassen and David Gushee, in Kingdom Ethics, explain:
Turning the other cheek has been misunderstood in Western culture that thought there were only two alternatives—violence or passivity. But since Gandhi and King, we can appreciate Jesus’ teaching better. In Jesus’ culture, “to be struck on the right cheek was to be given a hostile, back-handed insult” with the back of the right hand. In that culture, it was forbidden to touch or strike anyone with the left hand; the left hand was for dirty things (Stassen, Just Peacemaking, 64-65, 68-69). To turn the other cheek was to surprise the insulter, saying, nonviolently, “you are treating me as an unequal, but I need to be treated as an equal.” Jesus is saying: if you are slapped on the cheek of inferiority, turn the cheek of equal dignity (138-139).
As for very plausible cultural meanings of the 3 examples, Walter Wink's interpretation in his book Engaging the Powers: Discernment and Resistance in a World of Domination explained (emphasis mine):
At the time of Jesus, says Wink, striking backhand a person deemed to be of lower socioeconomic class was a means of asserting authority and dominance. If the persecuted person "turned the other cheek," the discipliner was faced with a dilemma: The left hand was used for unclean purposes, so a back-hand strike on the opposite cheek would not be performed. An alternative would be a slap with the open hand as a challenge or to punch the person, but this was seen as a statement of equality. Thus, by turning the other cheek, the persecuted was demanding equality.
Wink continues with an interpretation of handing over one's cloak in addition to one's tunic. The debtor has given the shirt off his back, a situation forbidden by Hebrew law as stated in Deuteronomy (24:10–13). By giving the lender the cloak as well, the debtor was reduced to nakedness. Wink notes that public nudity was viewed as bringing shame on the viewer, and not just the naked, as seen in Noah's case (Genesis 9:20–23).
Wink interprets the succeeding verse from the Sermon on the Mount as a method for making the oppressor break the law. The commonly invoked Roman law of Angaria allowed the Roman authorities to demand that inhabitants of occupied territories carry messages and equipment the distance of one mile post, but prohibited forcing an individual to go further than a single mile, at the risk of suffering disciplinary actions.[3] In this example, the nonviolent interpretation sees Jesus as placing criticism on an unjust and hated Roman law, as well as clarifying the teaching to extend beyond Jewish law.[4]
OT Precedent
It is also plausible that Jesus had Prov 25:21-22 in mind:
21 If your enemies are hungry, give them food to eat.
If they are thirsty, give them water to drink.
22 You will heap burning coals of shame on their heads,
and the Lord will reward you.
But who are our "enemies" in this verse? Does it apply to everyone? Very plausible real life example: how about when you found yourself to be a kindergarten teacher (who has a license to carry) in a mass-murder situation where the killer has been shown to be on a rampage to indiscriminately kill as many students as possible and who is now opening the door of your classroom full of 15 cowering 5 year old when the police has not come? Clearly, this is not the "enemy" Prov 25:21-22 is talking about, and neither is the "enemy" Jesus refers to in Matt 5:43. In killing a terrorist / mass murderer (as a last resort) you don't hate him, but you love the ones you are called to protect.