How did the society where the story of Shin Sekai Yori takes place come to be?
It's actually explained a bit at one point, though I can't remember the episode exactly.

Essentially, humanity got really violent and started destroying each other, probably via nuclear warfare or maybe previous people with supernatural powers. Shin Sekai Yori takes place on one of the remaining human colonies, where the elders are the only ones that know of the leftover science, knowledge from the previous era.

This explains why they're super paranoid about these kids with powers, because they're afraid that everyone will get destroyed again.