I vaguely remembered this from reading about it a while back but I googled it to check my recollection:
I'm by no means an expert, but my understanding is that the main radioactive product of nuclear fission (U-238) can be recycled to produce Pu-239 which acts similarly to the original U-235 used as a fuel source. This leaves only the fission products of U-235 to be dealt with which have a much more manageable half-life on the order of hundreds of years.
This is all coming from a site with a rather obvious agenda, so it would have to be double checked, but this matches with my recollection from other sources so I'm inclined to trust it.
Yeah, there are nuggets of truth in it, but it does not mention some stuff.
1. Nuclear fuel is not burnt down completely. The most common fuel used is U-235 (which I mentioned in my post above with the 700 million year half-life). It naturally occurs together with U-238. When the fuel is replaced in a nuclear reactor, usually about 4% are used up at that point, 96% are the original U-235 and U-238. U-238 has an even worse half-life of over 4 billion ears. As U-238 is not very useful in itself and U-235 is even more scarce than it initially was, it makes not much sense to enrich the U-235 still in there for productive use. So most of nuclear waste is actually the biggest part of the nuclear fuel.
2. Only a very small part of U-238 is turned into Plutonium, about 1%. Enriching it for commercial is a lot of expensive work. So if you want to work with Plutonium, you usually not use spent nuclear fuel (that also contains a lot of other isotope-byproducts as result of it's use which make work with it pretty unpredictable), you use special breeding reactors for the creation of Plutonium. But this is also problematic. First of all you have twice the nuclear waste: from the original breeding and then the fission products of the Plutonium-use. In reality it seems not very commercially viable, because mostly the resulting Plutonium stays unused. If that was less expensive, it probably would be done more often. But as it is, the produced Plutonium is mostly also nuclear waste.
3. There is an assortment of other fission products with wildly varying half-life and radioactivity. Besides that also the material used to encase the nuclear fuel also catches neutrons and turns into different isotopes or fissions into other elements. That means spent nuclear fuel contains a wild assortment of different radioactive and non-radioactive elements and isotopes, which is a difficult mess to extract anything useful from it.
Although most of the nuclear fuel is U-238 and in a nuclear reactor it is for months under a constant stream of neutrons only about 1% turn into Plutonium. So this thing described is happening, but not at all an solution for nuclear waste. As I wrote in my initial post, so far has nobody presented a convincing solution for nuclear waste. And I looked into it in the past. So this U-238 → Pu-239 (or Pu-240) works well on paper or a website, but the same people that present it as solution if asked about nuclear waste don't follow that route in reality as it is just too expensive and not commercially viable.