The waste from current nuclear power plants is highly radioactive and also toxic. It is stored on site under a pool of water until the worst of the radioactive isotopes have decayed but some of the particles will have half lives in the thousands to hundreds of thousands of years. This means the waste remains dangerous for thousands of years. Finding a place which is safe to store it is difficult, convincing locals to let you build that storage is even more difficult and then you have the cost of actually building long term storage.
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