Do people still visit it, and how long can you stay there?
The incident occurred in 1986 at reactor number 4 and yes Miguel while some areas within the area are being repopulated, there is an area known as the Chernobyl exclusion zone, stretching an area 2'600 kms from the power plant, cities such as Prypiat are now ghost towns and will never be inhabited again, the Exclusion Zone is one of the most radioactively contaminated areas in the world, scientific interest due to the high levels of radiation exposure in the environment, as well as increasing interest from tourists grows.
It occurred in 1986, and it is currently being repopulated so it seems that you are able to visit and not contract anything cancerous. Further, only 5% of the radioactive "components" (for lack of a better word) where released into the atmosphere. But, of course, this is only my hunch with what I assume to be are facts so I suggest digging a little deeper if this really interests you.
Here's if you want more specific details: http://funadvice.com/r/brt632181p8
BTW: Chernobyl is in Ukraine. Ukraine used to be a part of the USSR. Russia used to be a part of the USSR. So they were the same state when the catastrophe happened. They're not the same state now. Just sayin'...
Oh, sorry :/
Any time :)