Most toxic elements last forever. That includes the toxic elements in coal ash. Both coal ash and spent nuclear fuel need indefinite safe storage. It can be cheaper to store spent nuclear fuel because it's so much more compact than fossil fuel waste.

Remember that the status quo already has a large forever-waste cost of its own. Nuclear waste is a trade off vs. other kinds of waste, and it is cheaper than the status quo.

I mean, your car probably has a lead acid battery in it. Like uranium, lead is a toxic heavy metal, and it needs to be kept safe; it'll be as toxic in ten thousand years as it is now, and you should definitely avoid getting it in your drinking water (cf, Flint MI). Since your car has parts in it that need to be kept safe, in some form, forever, why is the cost of the car at the dealership not infinite? :)

Many, many industrial processes generate waste, sometimes horrible waste. Much of the waste needs to be stored in various ways, often indefinitely. Nuclear reactors aren't unique; it's more than a decent chunk of the population has noticed nuclear reactors, but is still oblivious to all the other costs of modern industrial lifestyles.

The good news about nuclear waste (and your car battery, come to that), is that dealing with it is actually fairly cheap. Would that we could say the same about the byproducts of burning fossil fuels!

Because there's not very much of it. The whole reason why nuclear is good is that fission of uranium has a much greater energy density than combustion of hydrocarbons.

And crucially this waste is solid and can be buried underground. Unlike carbon dioxide which is more or less impossible to store.

It’s solid waste. It just sits there and does nothing unless you get too close.

Because most reasonable people discount future costs, and those sufficiently distant to zero.