The age of a stone is tricky to determine, and often paleoscientists need to use multiple techniques to determine a rocks age. Dating will often be a multi step process involving determining the age of both the rock itself, and everything around it from soil sediments to artifacts found at the site.

In geocience, radiometric dating techniques often form the baisis for age measurements of rock, and work by measureing the relative abundance of radioactive isotopes of elements compared to the stable isotopes they decay to. If the isotope's half life is known, and the isotope it decays to is rare, scientists can determine how long it has been since a rocks formation from how much of the radioactive isotope has decayed to the more stable form. Techniques like Fission track dating of the damage trails left by the decay of unstable Thorium isotopes, U-Pb isotope dating of zircon crystals or glass or Potassium/Argon isotope dating are all examples of this technique and can determine the age of formation of a rock. (Lowe, D., 2011. Tephrochronology and it's application a review. In Quaternary Geochronology. V6 3-4 pp 423, Ch 9, has a good summary of these techniques)

The ages determined by these methods won't determine the age of a stone structure however, only the ages of the stones in the structure, which will only give an estimate of maximum age. This can be millions of years off depending on the type of rock. Electron spin resonance is a radiometric technique that can give an idea of when a surface was exposed if the stone is worked (Grun, R., 1997. Electron Spin Resonance Dating. In Chronometric Dating In Archeology vol 2), but these analytical techniques tend to be used in tandem with relative, and age equivalent dating techniques in archeological studies. These work by determining the ages of things around the structure (such as soil built up around the base of a stone, artifacts found on site, etc) to determine ages the structure can't be younger than.

Determining the age of organic matter in the soil built up around a stone since it was emplaced, or of human remains or artifacts found at the site can give a good estimate for the age of a stone. This can be done reliably using Radiometric dating of C14. Another good example of age equivalent dating is tephrochronology, dating using volcanic deposits. As volcanic material is emplaced in a wide radius over the course of a few weeks at most, if you radiometrically date the age of volcanic ash overlaying a stone structure you can use that to get a date the rocks cant have been emplaced after. The same can be done for ash layers under the stone get estimates of maximum age (Lowe 2011). If buildup of sediments around a stone is consistant or seasonal, you can estimate how long a stone has been in one place by how much sediment has built up around it, though this can be unreliable over very long timescales. A combination of deposition modeling, and of relative dating of prior and recent structures and deposits can then be used to establish a range of dates between which the stone must have been emplaced, and to estimate a precise date within that range for when the rock was emplaced.

Hope this answers your question!