Maybe focus on how the likely alternatives are worse. The world isn't going to run completely on renewables, and coal puts surefire poison AND radioactive isotopes into the air.
That is what safe looks like, nobody said perfect. It's like how cars are death traps but people can still reasonably call them "safe". You could do the same thing for solar panels but for the fact it isn't newsworthy if there is some toxic sludge in some dump somewhere is associated with solar panels.
There might not be any material that we produce on an industrial scale that doesn't leak hazardous or toxic chemicals. I looked up concrete [0] to find out what it's problem was, and it turns out concrete leaks toxic and radioactive materials into the environment because turns out natural rock is sometimes radioactive. Concrete is nonetheless pretty safe stuff.
[0] https://en.wikipedia.org/wiki/Environmental_impact_of_concre...
Would "they" rather we just dumped the nuclear waste off in some corner somewhere? Then it'd be a slow and certain leak too. If they want slow and certain rather than probably-nothing then that can be arranged. It'd still be less damaging than business as usual.
I mean, I get that people are scared but they are scared of their own imaginations as opposed to anything that can be engineered. I still think I'd be much better off having lived through a Class 7 nuclear disaster rather than my actual experience of spending years near a coal mine. Policy made for and by people who just live in fear for no obvious reason is never going to work. They're still going to be scared whatever the actual policy is and the rest of us are going to be forced to do stupid things in the real world.
A great perk of nuclear waste s that it's so small compared to the power produced, it's way easier to dispose and manage, than, say, expired wind turbines (which aren't recycled currently and take a massive amount of space).
https://www.kpluss.com/en-us/our-business-products/waste-man...
For example Herfa-Neurode underground repository contains (as of 2025):
690,000 tons of waste containing dioxins and furans , 220,000 tons of waste containing mercury, 127,000 tons of waste containing cyanide, and 83,000 tons of toxic waste containing arsenic.
https://de.wikipedia.org/wiki/Untertagedeponie_Herfa-Neurode
So nuclear waste is not a technical problem, mostly political problem.
And it's not very much talk about, but the real issue is geopolitics. Each nuclear trans-uranium waste repository (spend nuclear fuel) could in future be mined to retrieve the stored plutonium for nuclear weapons. And even better, over centuries the most radioactive short lived isotopes are transmuted into more stable isotopes, so the mining and handling of this material is much easier. And even even better over millennia, the undesirable isotope Plutonium-240 (with half-life 6561 year, much shorter then Plutionium-239 half-life 24110 years) will decay away and the reactor-grade Plutonium waste will itself transform, without any external action, into very usable weapon-grade Plutonium.
Therefor spend nuclear fuel and storage of spend nuclear fuel in non-weapon countries is subject to monitoring by the enhanced verification measures of the International Atomic Energy Agency (IAEA) Additional Protocol.
https://scienceandglobalsecurity.org/archive/sgs07lyman.pdf
https://en.wikipedia.org/wiki/Nuclear_proliferation#Addition...
https://en.wikipedia.org/wiki/Reactor_grade_plutonium
https://en.wikipedia.org/wiki/Weapons-grade_plutonium
U.S. goes to great lengths to prevent other countries from acquiring material usable for nuclear weapons. Like, for example retrieving plutonium and highly enriched uranium from Semipalatinsk Test Site in Kazakhstan.
https://www.belfercenter.org/publication/plutonium-mountain-...
"The Megatons to Megawatts Program, also called the United States-Russia Highly Enriched Uranium Purchase Agreement, was an agreement between Russia and the United States whereby Russia converted 500 metric tons of "excess" weapons-grade uranium (enough for 20,000 warheads) into 15,000 metric tons of low enriched uranium, which was purchased by the US for use in its commercial nuclear power plants."
"The program was credited for being one of the most successful disarmament programs in history, but its low set price for nuclear fuel caused Western companies to not invest in uranium refining capacity, resulting by 2022 in Russia's government-owned Rosatom becoming the supplier of about 50% of the world's enriched uranium, and 25% of the nuclear fuel used in the US."
However, nuclear energy is the safest form of electricity generation we have.
Because no technology is 100% safe.