Actually you can reprocess fuel rods. Typical rods are ~5% U235 at start with the rest U238, and over time most of the U235 gets burned up and turned into fission products. But you still often end up with ~1% U235 and some other stuff like plutonium or other actinides that could get burned up if it weren't for the fission products poisoning the reactor.
Japan and France reprocess the fuel, stripping out all the fission products but keeping the actinides (U235 and U238 and other fertile/fissile isotopes) then adding back in U235 and U238 to fill it out to ~5% fissile and 95% fertile material again. Then you put the rod back in a reactor.
This isn't as efficient a use of natural uranium as a breeder but it does make it the case that the only actual radioactive material you have to deal with waste-wise is the fission products. Everything else keeps going back into a reactor until it's burned up, or is U238 which is harmless (it's 99.3% of what we pull out of the ground and so could just get put back into the ground).
The fission products are nasty but that also means they decay very fast and become totally harmless (ie less radioactive than the original uranium ore) in ~300 years. We can embed them in a glass block, which renders them inert chemically and pretty impervious to leaching if exposed to water, etc. It's then easy to manage them until they're harmless (you can do whatever you want with them; the volume of material is not very high, and it's quite easy to shield against it).
France, for instance, can store the fission products from 1 tonne of spent fuel in 110L of glass, (see https://inis.iaea.org/collection/NCLCollectionStore/_Public/...) which corresponds in modern reactors (45GWd/tonne of spent fuel) to ~10GWh of thermal energy produced per liter of vitrified glass. By comparison, gasoline has an energy density of ~8.5kWh/L, so ~1.2 million times higher for the glass waste than unburned gasoline (coal etc is similar).
If the US ran 100% on nuclear for all energy (needing maybe 3TWe, 10TWh-th) and reprocessed our spent fuel, we'd produce around 24m^3 of vitrified glass per day, and its maximal volume would be ~2.7e6 m^3, or roughly 1 square mile one meter deep, to run the entirety of the US forever with nuclear power (that's 300 years of production, by the end of which the fission products are less radioactive than the original ore, so we can just dump any 300-year-old glass in the ocean or something and it's totally fine, can't harm anything).
Is that a lot of waste? I guess. But we burn that much gasoline in the US every 2 days, and the products of that just get dumped straight into the air. This stuff is solid and just sits there not doing much of anything except make some heat, then becomes harmless glass we can dump wherever. Pretty good deal imo.