This was the first question I asked when brought in my high school chemistry class a decade ago. Apparently there is too high of risk for a dirty bomb going off.

Storing it in the desert (or on site) is many orders of magnitude safer than shooting it into space. Not to mention the massive energy requirements for launching items into the sun. In fact it takes far more energy to reach the sun than it takes to escape the solar system entirely.

The simple fact here is that storage of nuclear waste is not a practical or safety issue at all, it's a political issue. If we can shut down the NIMBYS and irrational anti-nuclear activists this issue will be solved forever.

We don't need innovative solutions on handling the waste (that's already a solved issue), we need innovate solutions on getting political support for it.

People are way overly scared of nuclear waste. Though not as concentrated, radioactive materials exists in nature already. There are natural beaches you can sit on that will get a geiger counter going. If nuclear waste is stored reasonably it is not particularly dangerous. Especially when compared to all the other waste we dump into our air, water. I'd much rather have nuclear waste stored in my state than have the equivalent amount of coal smoke dumped into the air.

It's like how people are worried about windmills killing birds, even though glass buildings, cars, and house cats kill more and nobody ever cared about that.