We had geiger counters at all time on us. The most radioactive areas were a buried village, the sand under the douga that was used to wash it, and a can filled with garbage in Pripyat (which was by far the most radioactive things). Probably some kind of trap? Or a form of sculpture?
Yeah, you can definitely do it safely with the right gear and some planning. It’s both less risky on average and more tricky than most people think. Wouldn’t recommend drinking the water if you can avoid it though.
As far as the garbage. Hell if I know.
I was with an organized tour so it was relatively safe. I wouldn't go alone without proper training and equipment though. For the water, we drunk some "local" water at the inn. I think/hope that it's treated properly. Most of the food we ate was also bought by us before the trip.
So it’s safe … until it isn’t.