Rare coincidences make the technique feasible
Carbon-dating is a technique we take for granted in the modern archaeological era, but it depends on a couple of very serendipitous effects.
Basically carbon naturally occurs as three isotopes – C12, C13 and C14. However the C14 isotope is radioactive and so decays over time, with a half-life of 5,730 years. Carbon-dating simply measures the remaining C14 content in a sample to estimate how long the sample has existed since….
And there is the first piece of serendipity. We need to be able to know the starting C14 concentration.
Cosmic rays create C14 equilibrium
It turns out that there is a continual bombardment of the upper atmosphere with cosmic rays which creates new C14 from atmospheric nitrogen. That C14 then reacts with atmospheric oxygen to produce carbon dioxide which gets fixed into plant material by photosynthesis, which then gets eaten by animals, so that all living things contain an equilibrium, fairly constant concentration of the C14 isotope of carbon.
Once the living matter dies however, that concentration of C14 starts to decrease, by radioactive decay – hence it is possible to know how long any piece of once-living matter has been dead, by measuring the remaining concentration of C14 in it.
However there is another twist. That particular type of radioactive decay, a neutron into a proton, usually occurs within a day, for almost all other atoms. But it is the unusual long half-life of C14 which makes the technique useable.
Quantum interference of decay modes
The reason that C14 decays about 2 million times more slowly is that, according to quantum mechanics, multiple versions of the nucleus exist simultaneously, each having different amounts of angular momentum. And two of those particular versions are both trying to decay, but when considered as quantum waves which can interfere with each other, those two decay processes almost perfectly cancel each other out. Hence a very low probability of decay, and a long half-life.
And we could not have used just any other element – life is carbon-based, maybe oxygen or hydrogen would have worked if we were lucky. The only other element with a similarly decaying half-life is Beryllium 10 – not much of that around in living tissue fortunately – it’s toxic.
So, in summary, it just happens that the element that life is based on, has an isotope with a rare decay profile, and that isotope just happens to be constantly equilibrated by cosmic rays – archaeologists are blessed indeed.