The second term to define is 'Disposal'. In normal terms of human waste, this means stashing it somewhere, until it can turn into a nice compost, and feed a forest. It might have heavy metals, bacteria and such, but the trees can take care of it. For radioactive materials, it is the same principle -- stash it away until the next exposure to humans isn't any worse than the million other things that vex us. Those who have extreme views spread that to mean until every atom of hot Plutonium turns into lead, which isn't that great a thing. The rational view is to wait until the waste decays down to the level natural uranium ore, and cosmic rays. We humanoids have been exposed to radiation for 7 million years, and have evolved ways to deal with it.
-look up effects of radiation, look up natural sources, look up radiation release by burning coal.
As with all new things, the history of radiation disposal has many horror stories. The first was Radium, which allowed you to see your watch at night. This was eaten by the dial makers, and thrown out in the back ditch. It had dire consequences.
-look up Radium contamination.
Enough people got sick that we knew we had a problem. Then came the second thing about radioactivity -- it is so easy to detect, down to the smallest amounts. Try that with lead! By these sensitive detection methods, we found that all the radioactive material that we dumped in the back woods was coming back at us! The radionuclides loved to travel in the groundwater, animals dug it up and pooped radioactivity everywhere.
-look up Hanford, radioactive groundwater plumes, Geiger Counter, glowing bunnies.
And remember, we are confining ourselves to Radioactive Waste, not contamination, such as with atomic bombs. Contamination is something that has to be cleaned up, disposal is what to do with it after it is cleaned up.