Here the answer is 'sometimes'. There are several tricks to find the age of a rock -- but there are several caveats well: Rocks don't all have the same ways of coming into existence, some are formed via sediment, some directly from volcanism and some from repeated sedimentation, crystal growth, pressure and temperature changes, etc.
For a rock formed from sediment, age can often be judged by the collection of fossils and micro-fossils in the rock. Additional information can be judged by the radioactive decay of elements which are common in some rocks and whose by-products can't escape the stone.
For example, Models of the atmosphere and of natural isotope sources can be used to estimate the ratio of Carbon-14 versus Carbon-12 in recent (ice age Eocene) fossils. Since the decay rate is known for Carbon-14 (about 26,000 years half-life), by measuring a carefully selected (unexposed to the elements) portion of the fossil and determining the isotopic abundance of Carbon, one can tell how long the fossil has been protected from interchange of Carbon with the environment. Over longer spans of time, other radioactive species suffice such as potassium->argon, Uranium/Thorium etc.
The success of these techniques depends on the degree to which subsequent environmental effects can be ruled out. For example, some early results in seismology were invalidated we the effects of high than expected ground water solubilities were found. However, in dry climates, and in hard rock, the precision can be quite good. I think the dating of the anomalous Iridium layer world-wide has been measured to 65+- 1/2 million years by several people using independent techniques. (This layer is thought to have been caused by impact of an small asteroid -- which coincided with the end of the dinosaurs.)
There are other techniques, based on chemical reactions which are very slow or which can happen only is special cases, but these are not as dependable as radio-isotope dating simply because they are more sensitive to the environment.Finally, rocks on the surface i.e. rocks on the moon can be dated by weathering from the solar wind and cosmic ray flux, both of which are well studied and thought to be relatively constant for the last few billion years. This technique was used in some lunar samples brought back by astronauts.
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