Most of it is slightly radioactive stuff that you wouldn't want to be around long term but you could stand next to without issue. The problem with it is that you don't want it getting into groundwater/,/food chain etc, because then you drink/consume it, it gets stuck in you and you are 'standing next to it' for the rest of your life.

There is more dangerous waste too, but less of it. That offers the same problem, but worse. Normally, this is actively managed, but obviously not in this case.

The fuel itself (for nuclear ractiors, not relevant for weapons) can be recycled, but it's generally cheaper to get fresh fuel. Additionally, the processes are generally similar to nuclear weapons production, so people start getting nervous.