I visited CNPP about a year ago, shortly before they put the new enclosure in place. No problems wandering around - I walked up to within a couple of hundred metres of the reactor (the highest background levels you'll experience unless you go looking for fragments of the core, which we did find). Providing you don't lick the surfaces there's little issue radiation-wise though it would be smart to take a personal dosemeter with you to be aware of hot spots if you go walkabout (virtually anyone offering to take you into the zone will rent you one). I did choose to go during a wet season though to help keep the dust down (construction lorries kick up a disturbing amount of material from the road). Great electrical storm whilst we were there - one day one of those will ignite the red forest and then we will all be in trouble again.
If you keep out of places like the hospital basement then the biggest risk in Pripyat is falling bits of building, floors giving way, broken glass, trip hazards and asbestos (also in the opinion of my guide who was evacuated from there the night of the disaster, aged 6, never saw her father again). Officially no one is supposed to enter any of the buildings anymore but this was clearly being ignored last year.
Also climbed the duga. Guards there were being a little awkward but let us on site eventually. Guide was worried that someone might take a pot shot at one of us once up there. My tip is to not follow someone else up the ladders so you don't have to ingest whatever detritus is showering down on you from their boots.