You need special permission to go to Prypiat and everyone inside the exclusion zone wears film badges to monitor their dose but in general it is pretty safe around there nowadays.
There are several thousand people working on site every day so its not really quite as most people imagine it to be. Obviously the villages around in the exclusion zone, the forest etc are not the nicest places in the world to be but there is loads of wildlife thriving there. We would see elk, wild boar, eagles, turtles, snakes and many many other things almost daily commuting back and forth to site.
Really interesting place to be.