For years I wanted to visit Chernobyl Exclusion Zone - the disaster played an important role for our generation, became part of the popular culture, now being both a nostalgic view of 1986 Soviet state of the art as well as a place where you can see how quickly nature can reclaim man-made structures.
You must take a tour to visit Chernobyl, and you must register for it at least 2 weeks in advance with correct passport details - this is required by the government. Day tour may be enough with very early start from Kiev and late return back. But on the tour you realize that having more time there would be even better. Take a 2 day tour if you have time. I used Chernobyl Zone, with whom I got a less crowded tour with a group of only 6 people and 2 guides.
Average radiation levels there are not dangerous nowadays, they say that in a day you get the same amount as on a trans-Atlantic flight, provided that you don't stick your nose to the wrong place, like Ruby Forest, basements or other lower areas. The guides carry radiation dosimeters, and will tell you if going somewhere is not recommended. There are many people working in 2-week shifts in the exclusion zone, and they are alive. We even had an "ecologically-clean" lunch in the canteen right next to the power plant. On the way there and back you will also pass several checkpoints, checking for any radioactive dust that may have been caught by your clothes. I was fine.
Besides seeing the reactor from a distance and visiting the World-famous abandoned city of Pripyat, founded in 1976 to show-off modern Soviet special-purpose city design, you will also see the less famous, but really cool Duga Radar (aka Chernobyl-2 or Russian Woodpecker), which purpose was to detect US missiles flying to attack the USSR during the Cold War.
The Chernobyl town itself is located a bit further away from the reactor than Pripyat and, despite being also mostly abandoned, is nowadays home to some working shops and hotels for the workers of the zone.
Walking in the zone also constantly reminded me of 1979 Stalker movie, filmed in and near my home city of Tallinn, but amazingly predicting the existence and current mood of the Exclusion Zone.