As the name suggests it boils down to digging a very deep hole - over half a kilometre down in the case of Onkalo - and sticking the waste into it. They then backfill the tunnels and simply leave it in there to decay. This particular facility will open around 2020 and continue to take waste for at least a hundred years. The engineering of the thing is Herculean, not only due to its size, but because it has to last for at least 100,000 years, which is roughly the time that it will take for most of the most dangerous and long-lived nuclear waste to become safe. Most of the world's nuclear waste is currently stored above ground, something which is dangerous for obvious reasons, so the benefits of locking it away underground, safe from natural disasters and human interference are manifold.
One of the most interesting aspects of the documentary was the debate being had about whether or not to leave any indication that the facility was there. The argument was that if it was going to last for a hundred millennia it would probably outlive human civilization in its current form, and quite possibly outlive the human race full stop, and, as such, should they leave warnings to the future imparting the immense danger within the facility, or just demolish the above-ground structures and hope that it would be forgotten.
The ideas that the people in the "leave a warning" camp were coming up with were fascinating. The major idea was to leave obelisks, or even a small "library" of sorts, carved with warnings not only in all of the world's languages but also in pictographic form, should those who discover it have regressed to stone-age illiteracy, or even not be human altogether. Some of the other ideas were much more abstract, like creating nightmarish architecture such as a landscape of thorns or huge, threatening monoliths, or even a depiction of Munch's The Scream to warn people to stay away. The other side of the argument stated that things such as these would just pique the interest of some, particularly if they had retained a folk memory of a huge underground cavity filled with something worth keeping secret, and that they should just eliminate all trace of the facility above ground.
The main subject of the Sunday Times' article was the current push to build one of these things in the UK, although they are understandably having difficulty in finding a site. All of Britain's nuclear waste will be stored in it, and that doesn't just mean spent fuel and their by-products, but everything ever associated with nuclear power from the materials used to construct reactors to the protective clothes worn by workers at the plants.
At the moment all of Britain's waste goes to Sellafield. Some of it is reprocessed, but a lot of it prepared for long-term storage. They do this by vitrifying it, which makes it very stable and is an apparently irreversible process. The article described the vitrification room as "the most dangerous enclosed space in Britain". The work is (obviously) done by robots controlled by scientists protected by many feet of lead glass and reinforced concrete walls. The article claimed that if there were even the tiniest crack in this wall anyone walking past it would die instantly.
I don't really know what to make of it all really. I was tentatively pro-nuclear fission, at least in the short-term, but discovering exactly how gargantuan a task it is to deal with the waste has made me somewhat uneasy about building a new generation of nuclear power stations. Not that I have any answers as to how we are going to plug the energy gap otherwise.
The documentary in question: Nuclear Eternity.
A Guardian article on the documentary.
Wikipedia entry on nuclear waste.
Unfortunately I cannot link to the Sunday Times' article because of the paywall, which is a shame because it was highly informative, much more so than the Guardian article linked above.
Apologies for the massive essay; I didn't realise I had written that much. I suppose it shows how intriguing I have been finding the subject.