Disposing of nuclear waste is actually a big deal in the US. Last I heard, nuclear waste is kept on-site or close by since the Yucca mountain storage facility was never built[1]. These storage facilities weren't built or meant for long-term storage and could potentially (likely?) have radiation leaks.

edited to add links

[1] https://www.scientificamerican.com/article/what-does-the-us-...

[2] https://www.forbes.com/sites/jeffmcmahon/2019/05/31/new-map-...