Chernobyl
From what I understand, it's all the cold war era, bomb related stuff that's the problem, from the fire and other activity around that time. Rather than come up with a proper solution for the waste material they literally stuck everything in "storage ponds" - massive concrete lined pools that they just poured the dodgy, left over material into cause they didn't know what to do with. Now the pools are literally falling apart cause of age and being battered by ionising radiation for 75yrs, and the guys working there now are having to fish out this stuff and store it away properly without breaking anything else in the process
Plus they've no idea what's actually in there in some cases
The Sellafield era stuff is far more controlled - I visited the site with University a number of years back and it was amazing how controlled it is. In the high level waste storage building, where the vitrified material was held, the radiation level was lower than background for the area. I heard from a colleague who worked in a radioactivity lab during Chernobyl (in NL) that during the Chernobyl event they were legally required to store the rain water for a few days, because they couldn't release that level of radioactive terracore water.
but all the rest of the place it just rains and nobody's looking.