A nuclear EMP would cause prolonged grid failure. This in turn would cause famine and widespread violence. Wandering gangs of thugs would methodically plunder homes. They would have the advantage of superior numbers, of experience, and of surprise. Normally the police deal with such gangs, but police are likely to stay home with their hungry and defenseless families, and are too few in any case.
Neighbors could join together to defend themselves, but unless they planned ahead, they would have less food and fewer weapons than the gang members. At best, they might achieve parity, a kind of “symmetric warfare” in which for every bad guy killed, one good guy dies. These are not very attractive odds if there are lots of bad guys.
The best solution I’ve come up with is for people to camp out on farms and defend them, using defensive walls and open fields of fire to improve the odds. Besides, if farms are ruined, everyone will starve eventually, no matter how much food they have stored.
Afterthought – Symmetric warfare is scary to bad guys too. They would rather have the advantage, and if a neighborhood puts up a brave defense, they may well retreat and move on to easier prey.