Deep thoughts

How exactly do you measure a half life that is > 10^25 years? Very very carefully.
(a) You go deep underground, so as to stop Cosmic Rays masquerading as the decay products you're looking for in your detector. Try 8000' under rock in South Dakota. That oughta do it.
(b) You use ultra-high-purity materials: materials with almost no radioactivity of their own, to stop those decay products from looking like the decay your looking for. Try materials that are literally a billion times freer of Uranium and Thorium than the dust in our offices. That might be good enough.
(c) Try collecting a tonne of germanium. Then you might have enough nuclei in there that a few will decay in the way your interested in. And for good measure, you can use the germanium you've collected as a solid-state detector too. Fortunately Germanium will work well in that regard. If you can get enough. And pay enough to the Russians to purify it for you.
(d) Wait several years. At least you know that the longer you wait the better lower bound you'll set on neutrinoless double-beta decay.

So if you do all that you might, just might, see neutrinoless double-beta decay. (Something to think about: how does your chance of seeing NDBD depend on the different things you do in (a)-(d)?)

But neutrinoless double-beta decay---if seen---would be a revelation. It would tell us that neutrinos are Majorana particles. They are their own anti-particle. Personally I think that'd be worth a Nobel prize.

And the rate of neutrinoless double-beta decay is proportional to the square of a linear combination of neutrino masses. So if we see it we learn something concrete regarding the absolute mass scale of the neutrinos: a quantity for which we at present only have upper bounds. (From cosmology and from studies of the end point in tritium beta decay.) Indeed, even if the next generation of neutrinoless double-beta decay experiments sees NOTHING that will still pretty much rule out the "inverted hierarchy" of neutrino masses. It's good when seeing nothing still gets you something.

But to get anything in terms of physics results will take a long time. In the meantime we learn that everything is radioactive. Dust, plastic, air. Everything.

You know one experiment of this type---one based in Europe---had a lot of background events coming from the surface of their crystals. The lines indicated Polonium-210 contamination on the surfaces. I heard today that people think that this background comes about because the people handling the crystals are smokers. and cigarettes are full of polonium. So smoking really is bad for your health: you get lots of radioactivtity on you as a result.

But when a neutrinoless double-beta decay guy says "lots of radioactivity" he probably means 1 decay per kilogram per 20 days or something like that. Because these guys are the ultimate purists. You have to be when you want to find something that only happens once a year in one nucleus out of all the nuclei in a couple of hundred kilograms of stuff. Like I said, delayed gratification. Big time.