Due to the energy density of nuclear fuel, the spent fuel (waste) is also in very small volumes. For example, all the spent fuel ever generated from nuclear power plants in the US could fit inside a Walmart. The UK has a plan for long-term disposal of high-level nuclear wastes in an underground system of tunnels and chambers. The organisation responsible is called Radioactive Waste Management Ltd. They are yet to choose the location. Waste is safely stored above ground in the meantime.
Interestingly, there doesn’t seem to be any plan to dispose of the waste from solar panels and wind turbines, which both can include toxic chemicals, or to recycle the steel and other useful materials. Solar and wind consume much more steel, concrete, glass and rare earths than nuclear, and therefore have larger mining impacts (again, more land).