What does it mean when an isotope is stable?

Does stable mean that an isotope has a very long half life... or does it mean that fissure is theoretically not possible, or does it mean that the isotope has a very long half life, but the exact number is unknown?

"Stable" effectively means that there is no experimental evidence that it decays. However, there are nuances within that statement.

  1. Most of the "stable" light nuclei can also be shown to be theoretically stable. Such nuclei would have to absorb energy to decay via any of the known decay modes, and so such decay cannot happen spontaneously.

  2. Many heavier nuclei are energetically stable to most known decay modes (alpha, beta, double beta, etc.) but could potentially release energy via spontaneous fission. However, they have never been observed to do so; so for all practical purposes they are considered stable.

  3. Some nuclei could potentially release energy via emission of small particles (alpha, beta, etc.), but have never actually been observed to do so. Such nuclei are often called "observationally stable".

  4. Several nuclides are radioactive, but have half-lives so long that they don't decay significantly over the age of the Earth. These are the radioactive primordial nuclides; your example of xenon-124 is one of them.

Note that nuclides can in principle be moved from categories 2 or 3 into category 4 via experimental observations. For example, bismuth was long thought to be the heaviest element with a stable isotope. However, in 2003, its lone primordial isotope (bismuth-209) was observed to decay via alpha emission, with a half-life of $\approx 10^{19}$ years.

One could defensibly claim that the nuclei in categories 2 & 3 are radioactive but their half-life is unknown; after all, the totalitarian principle says that any quantum-mechanical process that is not forbidden is compulsory. If you want to take this perspective, though, you have to assume that we have a good enough grasp on nuclear physics to know what is forbidden or not.

This half-life of $1.8\cdot 10^{22}$ years was actually measured. At first glance it seems impossible to measure such a long half-life. But let's go through the numbers to see that is indeed scarcely measurable.

The actual measurement has been done with the XENON1T detector. This experiment used 3 tons of liquid xenon, which are around $10^{28}$ xenon atoms. Natural xenon is known to contain $0.1$ % of the isotope xenon-124. So they had around $10^{25}$ xenon-124 atoms. The experiment detected a few xenon-124 atoms per day decaying to tellurium-124 by double electron capture (see "Dark-matter detector observes exotic nuclear decay"). Now you can use $\frac{dN}{dt}=-\frac{N}{t_{1/2}}\ln(2)$, and find the half-life of xenon-124 to be $t_{1/2}=1.8\cdot 10^{22}$ years.