There are any problems with fission that are all related to the extraordinary danger of handling the fuel, byproducts, and the sites themselves.
The cost of them is huge, some people are hoping that modularity will help with construction, but it is still astonishingly expensive.
The problems of handling the fuel has been solved, in theory and practise. Except when commerce is involved. When the money people get involved corners will get cut, and we are back to incredible danger. Technically solvable, but I would not go near it. I have known too many business people.
The problem of the long-term waste is entirely beyond us. There has been no practical progress on this front. Long term waste (including some parts of the assemblies themselves) are very dangerous for hundreds of thousands of years.
This is, with current technology that can be bought to bear, unsolvable.
The only thing we can do is put it in a stable site, be ready to move it when the site becomes unstable (nowhere on Earth is known to be stable on such time scales), and find a way of communication, across thousands of generations, just how poisonous this stuff is.
Maybe our ancestors will get lucky and find a way to safely dispose of it....
So fission power is making future generations pay for today's consumption.
Fortunately for us it is moot. The costs of renewables is dropped to the point that the only reason for fission is to build the capacity for nuclear weapons.