If Starship flies as much and is as safe as SpaceX predicts, they will have enough data to show empirically that Starships launches have an expected risk below 0.1% (i.e., it won't rely on theoretical models). That 0.1% isn't nothing, but it could easily be acceptable if the nuclear waste disposal Starship flight is launched from a sea platform (which is currently a planned Starship capability) in the middle of the Pacific. For instance, it's already been well argued that dilution of radioactive waste in the oceans could also provide a good disposal option, leading to a worldwide increase in background radiation that is much smaller than the baseline levels.

(There are a lot of important details here, including (1) the disputed argument that some isotopes might be preferentially concentrated by some organisms to high levels in any dilution scenario and (2) that an explosion of Starship in the Pacific would not dilute the radioactive waste as uniformly as a dedicated waste dilution program. My comment is just trying to point out that this could plausibly be acceptably safe and more careful analysis would be needed to check.)

Could we not put it in a thick lead box which is designed to survive explosions?

Why don't they just make the whole plane out of the same material as the black box?

Thicker box = more weight = $$$. Plus is anything really resistant to thousands of tons of rocket fuel?

Sure, more weight but at the bargain price of $100/kg

Could it be resilient enough? I have no idea. I'm a programmer.

We already make containers that can tolerate being pummeled head on by freight trains without leaking, so I imagine that we could definitely engineer something similar for space travel.

Isn't Uranium as sturdy as any material you'd pick to encase it in?