Vitrification is the process of turning the spent fuel into a glass-like substance. This includes chemically stabilising the fuel so that it's not chemically reactive, and immobilising it in a crystalline structure. While vitrified waste is stored in containers, it doesn't really need them. The only way it would be a health hazard is if it were liquefied with sulphuric acid or aerosolised by explosion and then ingested.

Even if all that were to happen in the worst possible case scenario (as has happened at Lake Karachay in Russia where untreated nuclear waste leaked into the groundwater in large volumes), it's still a long shot from being as deadly as the fossil fuels we take as a given today. That sets the upper worst hypothetical imaginary case for vitrified nuclear waste if some super volcano erupts at the storage site and converts it back into a chemically active form.

You can buy kitchen plates and glasses made from uranium glass, and that stuff is relatively a bigger hazard than vitrified nuclear waste. So is the americium in your smoke alarm, so are the various nuclear materials used at your local hospital for radiography and radiology.

To really nail this point home: coal fired power plants release more radioactive waste into the environment that nuclear plants do. The trace uranium found in coal can't be combusted and makes its way out the chimney, and that is more than a nuclear plant releases including disposal [1].

[1] https://www.scientificamerican.com/article/coal-ash-is-more-...