Regarding nuclear fission energy people often talk about reactor accidents. In my opinion (and that of many others) the far bigger and yet unsolved problem is nuclear waste. Uranium-235 has a half-life of over 700 million years. We have no way to store stuff for that amount of time. I have seen no sound solution to this yet.
One of the main problems here is that everyone goes immediately NIMBY and BANANAS when it comes to nuclear waste.
The basic idea is to bury the nuclear as close as possible to the mantle (aka as deep as possible), and doing so in a geological inactive region. Add to this the aforementioned nimby and bananas and the options as to where to build such a facility are severely limited. There are several in existance, but not many active ones that are considered final repositories. There are however several getting built, like this one in Finland that I know of for instance.
The problem is less that we don't have the structures (bury deep enough and earth itself will take care of it via tectonics), but more that language evolves over time. How to make sure that explorers from, let's say, 1000 years in the future will understand what is written on the storage facility - and how to ensure that they don't dismiss the warnings as some old superstition? Because those two points are actually what's holding up the construction of new facilites right now.
One solution might be the deep borehole storage. Basically they would drill a hole about 5 kilometers deep, so way below anything that could affect the surface or ground water, put caskets with the waste and neutron-absorbing salts one over each other, seal it with a mile or two of concrete and fill up with rocks and finally dirt. Only problem is that you'll need several of those since their storage capacity is quite low. But their location should ensure that nobody goes to check there or try to dig them out - it's just to deep and hard to reach.
Another solution could be to put them into a rocket end shoot them into the sun. But if the rocket fails...
Finally, one way that's getting researched is nuclear transmutation. The basic idea is to transform the radioactive isotopes with ß-radiation (which are basically the cores from Helium without it's electrodes) into heavier elements, but with stable isotopes. This is also how nature creates the heavier elements in supernovas.
Btw, you listed U-235, which is the main fuel of most nuclear reactors (there are exceptions that run on Plutonium or Thorium, but any reactor running on Uranium uses U-235). If there's some of it left, then the fuel rods will get reprocessed to use it, since only 0.72% of the Uranium is U-235 and thus relatively hard to get. It gets enriched to about 5% for usage in nuclear reactors (30-40% for nuclear weapons btw). No need to store that stuff, since it's exactly what we want for nuclear reactors.
Finally, by far not all nuclear waste is coming from nuclear reactors. The vast majority is actually coming from laboratories and hospitals (mostly from radiotherapy), but is only lightly radioactive. So even without nuclear energy, we would still need repositories for radioactive waste.
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