Nuclear waste is problematic because it emits radiation. It emits radiation because the atoms are unstable, and are decaying into other, more stable atoms. The rate of this decay is normally measured by the half-life - which is the time by which 50% of the element has changed into another, more stable element. The by-product of this decay is that it emits radiation, which is harmful to life. However, it emerges from this that the rate of the decay is directly proportional to the amount of harmful radiation that is emitted. Many things encountered in ordinary life are unstable to some degree or another, but ordinarily either the quantity is so low, or the rate of decay is so slow, that the amount of radiation we recieve under normal conditions isn't particularly harmful.
Now, nuclear waste. This falls into three broad categories. First is those with a half life of less than 50 years, like strontium-90 and cesium-137. These are horribly, horribly dangerous isotopes, and will do horrible things to you if you get near large quantities of them. However the fact that they are so very dangerous means that they are by turn, not going to be a threat for very long. After a short period of time (~30 years for the two I mentioned), they are half decayed already. Devising a storage system to store these sin't very challenging - the quantities are small, and they don't need to be stored for terribly long.
The second category is the one that the anti-nuclear lobby loves to blur with the first. These are isotopes like plutonium-239, with half lives in the tens or hundreds of thousands of years timelines (~24000 years for Pu-239). However what the anti-nuclear lobby fails to mention is that by the very fact that these elements have such a long half life, they are by definition not emitting all that much harmful radiation - many of the waste products in this category similar radiation per kg than coal ash. If we're happy storing coal ash behind nothing more than a chain-link fence, objecting to these waste products is sheer fear mongering, nothing else.
The third category is the problematic one. Here we have things like radium-226, which have half-lives in the thousand-year range (1300 years for ra-226), but are still radioactive enough you want a bit more safety than just piling them up behind a fence. However the good news here is that the quantities of radioactive products in this range are tiny. So tiny the entire non-military waste of these isotopes we, as a species, have ever produced would probably fit onto a single train. I think getting rid of one of the most effective ways of power production we've discovered to date for this amount of waste might, just maybe, be described as a colossal overreaction.
Sorry this post got a bit long!