In living organisms, which are always taking in carbon, the levels of carbon 14 likewise stay constant.
We can't date oil paints, because their oil is "old" carbon from petroleum. And third, it is common to soak new-found fossils in a preservative, such as shellac.
Carbon 14 is continually being created in the Earth's atmosphere by the interaction of nitrogen and gamma rays from outer space.
Since atmospheric carbon 14 arises at about the same rate that the atom decays, the Earth's levels of carbon 14 have remained constant.
The C14 will undergo radioactive decay, and after 5730 years, half of it will be gone. So, if we find such a body, the amount of C14 in it will tell us how long ago it was alive. The method doesn't work on things which didn't get their carbon from the air.
This leaves out aquatic creatures, since their carbon might (for example) come from dissolved carbonate rock.