The point of the “Lex Talionis” or the eye for an eye rule was not that it was meant to encourage harsh vengeance but that it was meant to limit and constrain revenge and keep it within the boundaries of justice. The alternative was for blood feuds that would run and run with ever escalating consequences.
However, Jesus in his desire to take our pursuit of righteousness further, pushes us beyond that. “Not just limited vengeance” he says. Rather, be willing to accept the unjust attacks and burdens of others.
I have heard it argued, quite persuasively that he is in fact encouraging a form of non-violent resistance here. The one likely to demand that you gave him your coat or carry his bag was likely to be your oppressor. Turning the cheek meant that they had to slap you with the back of the hand which was demeaning to them. Carrying their bag two miles might actually cause them problems with paperwork.
That all sounds brilliant until we realise two things. First of all that this is simply a way of softening the blow by speculating about what Jesus really meant but did not say. Secondly, it does not fit with the context here.
Remember that Jesus has talked about God’s happy people being those who hunger for justice/righteousness, who love mercy, make peace and themselves face crushing persecution so that others might consider them meek. It is the words of the Beatitudes that give us a better clue as to what Jesus was getting at.
Further, in the immediate context of the fulfilled law, we see that Jesus lifts our eyes away from the imperfect justice of human courts to the perfect justice of heaven’s courts. The point he is making is not that we should find clever ways to exact our vengeance and resist persecution. Rather, we are once again being reminded not to be so focused on the skin deep that we lose faith in God’s provision for our hearts, for our whole lives and for our eternal future.
Finally, the next paragraph makes clear what this is all about. We are to love our enemies.