It can be done experimentally, but it doesn’t require measurements of time as short as the half life. For the example you provide, it’s possible that it was done via theoretical calculation; I don’t really know for sure. My point was that it’s not necessary to use the half life as your time unit. Times between measurement of amounts of material can be significantly longer than the half life. For instance 50 half lives would result in a reduction by about 10^-15. With a good mass spectrometer and a large enough sample, this would be detectable. This would also allow experimental determination of half life.
This isn’t really my field, so this is only speculation, but it may also be possible to determine half lives of very unstable nuclides by taking advantage of special relativity. The half life is a proper time, which means a time measured by an observer in a comoving reference frame with the include. If a nuclide is accelerated to near light speed, it could theoretically be made to last quite a bit longer as measured by a laboratory clock. Particle accelerators are quite capable of accelerating a beam of He5 nuclei to such high speeds. There is no theoretical limit on the time dilation factor that can be achieved. A time dilation factor of even 10^6 or so would make the experimental measurement of the half life possible.
I have no dispute with you - but most of your remarks up till now, while scientifically sound, would be totally impractical for measuring the half-life of a substance as short as that of He-5.
With regards to your latest remarks - that one could observe the substance for a period of time many factors longer than its (short) half-life: That presupposes that one has a sufficient quantity of the substance in question - but the shorter the half-life of a substance, the more radioactive it is, and the more energy per unit of time it releases.
In the case of a substance with such an extremely short half-life as He-5, any amount of the substance that could be scraped together (but how?) even only approaching macroscopic quantities would be so volatile as to preclude experimental observation.
A good comparison would be the case of the determination of the half-lives of super-heavy transuranic elements, with Atomic Numbers of, say, 103 and above. They all have half-lives on the order of mere seconds, if I recall correctly (many isotopes with half-lives of only fractions of a second).
If I understand it correctly, scientists have never been able to produce more than a few atoms at a time of such transuranics - certainly never enough to actually take a measurable quantity and wait for several half-lives to see how much of the original sample has "survived" and thus to determine the half-life using the method you proposed in your last comment above.
Instead, they bombard macroscopic targets of heavy elements (like Uranium) with, say, Xenon nuclei in a cyclotron, and then observe the decay in cloud chambers. Every now and then, they observe the millimeter-long track of a nucleon (travelling at near light-speed - so relativistic effects like the one you mentioned do, indeed, have to be taken into consideration) in the cloud chamber, deduce its mass and charge (the cloud chamber has been placed in an artificial magnetic field), and calculate its lifetime.
My point is that no physicist has ever had anything like a macroscopic sample of these transuranides, and has never been able to, say, weigh it and then wait and watch (over multiple half-lives) as it diminishes in mass.
But this really isn't my field either, so feel free to correct me!
Regards,