Not so. Nuclear fuel isn't actually meaningfully radioactive - it's fissile, so when bombarded by neutrons it shatters (aka fissions), but leave it alone and it will decay so slowly that it emits essentially no radiation. Basically the longer the half-life, the less radioactive the material, and nuclear fuel has a very long half life - it has after all been present since long before our planet formed from the remnants of ancient supernovas. Uranium-235, the most common fuel, has a half-life of 700 million years. Plutonium-239, the other common fuel, has a half-life of "only" 24,000 years, but the banana you just ate is still far more radioactive than a warehouse full of freshly refined nuclear fuel.
Shatter it in a fission reactor though, and the resulting fragments tend to be very radioactive. Some are *extremely* radioactive, with half-lives in the range of hours to weeks, but store those for a few years and they pretty much completely decay into inert atoms, so they aren't a big problem. Dealing with those is why spent fuel typically has a "cooling off period" of a few years before being shipped away from the nuclear plant.
Most fragments though have half-lives in the range of a few months to many years - short enough to be dangerously radioactive, but long enough to present some challenges for storage - it will take many centuries to decay to safe levels. Still, bury it in a nice deep hole in good dry, stable ground, and it will probably decay to safe levels before it manages to escape. Something like Yucca Mountain would likely be well suited to this.
The REAL problem though is that current reactor designs only consume a few percent of the fuel, and then we go and store that highly radioactive waste still all mixed in with the 90+% of unused fuel. And that gives us the worst of both worlds - it's highly radioactive, and as the waste decays it releases a bunch of energetic neutrons which cause some of the surrounding fuel to fission, creating fresh new waste to replace the stuff that just went inert. The combination of waste and fuel will thus remain dangerously radioactive until all the fuel has been converted to waste, a process that will take tens to hundreds of thousands of years. And on those timescales there's no realistic way to contain it reliably, you're starting to get into geological timescales there, and there's no longer any such thing as stable ground.
THAT is why fuel reprocessing (or radically better reactors) are important - not just because it reduces the amount of waste, but because it simultaneously radically reduces the time it will take that waste to become inert, down to timescales where human ingenuity can at least hope to contain it.