https://en.wikipedia.org/wiki/Yucca_Mountain_nuclear_waste_r...
There is, we know what it is, but no one wants to pay for it - it's called deep borehole storage. You basically put your fuel in cylinders, dig boreholes few miles deep, put all the waste at the bottom, done. It's not coming back ever, it doesn't pose any danger to anything or anyone on anything other than geological timescales(and if the geology starts pushing stuff buried 5 miles deep to the surface you have other bigger problems to worry about).
It's estimated that "just" 800 boreholes would be enough to store all nuclear waste ever produced.
But yeah, cost is one thing, but the other is that this material then becomes truly and irreversibly irretrievable - which actually might not be desirable since we know that even spent nuclear fuel can be reprocessed to make more fuel or weapons, which is not something that countries like US would want to dismiss as a possibility.
And also - storage of spent nuclear fuel is actually a lot safer and less scary than most people think. In Netherlands they literally made a museum out of their spent fuel storage facility, you can walk in between the casks that hold material radioactive enough to kill you in minutes, it's that safe.
On the specific topic of reprocessing though. Reprocessing achieves 2 things: 1. it can extract usable fissile fuel from spent fuel, and 2. it can reduce the amount of long lived radioactive waste, by a factor of 30. Point 1 can be further split in 1.a. usable fuel for the current generation reactors and 1.b. usable fuel for future, fast reactors (U-238).
1.a. Per wikipedia [1]
Reprocessing the plutonium into usable fuel increases the energy derived from the original uranium by some 12%, and if the uranium-235 is also recycled by re-enrichment, this becomes about 20%
1.b. reprocessing in order to extract U-238 for fast reactors. That's a nice concept, but if we ever build fast reactors that can burn U-238 (fingers crossed), we already have a huge stockpile of depleted uranium. The US alone has more than half a million tons, and the rest of the world at least as much. That's enough to keep the lights on in the entire world for hundreds of years.
2. reprocessing in order to reduce the waste. That makes sense. But burying the waste is probably cheaper. We have already buried hundreds of thousands of tons of waste at WIPP. We know it works and it is safe. We know we will need to eventually bury some waste, even if its 30 times lower. If we, as a society, agree to open some deep geological repositories for nuclear waste, then it doesn't make all that much of a difference if we bury 10000 tons or a million tons.
[1] https://en.wikipedia.org/wiki/MOX_fuel
[2] https://en.wikipedia.org/wiki/Waste_Isolation_Pilot_Plant
https://en.wikipedia.org/wiki/BN-1200_reactor
In early 2012, Rosatom's Science and Technology Council approved the construction of a BN-1200 reactor at the Beloyarsk Nuclear Power Station. Technical design was scheduled for completion by 2013, and manufacture of equipment would start in 2014. Construction would begin in 2015 with first fuel loads in 2017 and full commercial operation as early as 2020.
...
In 2015, after several minor delays, problems at the recently completed BN-800 indicated a redesign of the fuel was needed. Construction of the BN-1200 was put on "indefinite hold", and Rosenergoatom stated that no decision to continue would be made before 2019. In January 2022, Rosatom announced that a pilot BN-1200M would be built by 2035.
It isn't a huge risk. The really hot stuff decays quickly and that can happen next to the power plants no problem. What you end up left with is far less energetic and is self-contained – look up dry casks. You can stand next to them no problem and they are incredibly sturdy.
That's ignoring reprocessing. If it is radioactive enough to cause serious harm long term, it is radioactive enough to still use as fuel. There are reactor designs that can re-use them.
Honestly, I would rather have some dry casks stored at Yucca Mountain than breathing coal dust (that's often radioactive).
But there's still no solution for safely storing or disposing of nuclear waste, so all those power plants currently have a stockpile of spent fuel rods, which is a huge risk imo. But everyone's like "not in my backyard", even though the backyard is a deep cave in a mountain or down in the earth where the stuff will be put in lead and reinforced concrete, the cave sealed off or collapsed, forgotten by time and where it will remain dormant and slowly go inert over the next geological era.