See below for details about the 45,000 annual varves in Lake Suigetsu. For example, polar bears that eat seals aren't getting their carbon from an atmospheric source.
It is produced in the upper atmosphere by radiation from the sun.
(Specifically, neutrons hit nitrogen-14 atoms and transmute them to carbon.) Land plants, such as trees, get their carbon from carbon dioxide in the air. The same is true of any creature that gets its carbon by eating such plants. Suppose such a creature dies, and the body is preserved.
If you hear of a living tree being dated as a thousand years old, that is not necessarily an example of an incorrect dating. Wood taken from the innermost ring really is as old as the tree. We can date things for which historians know a "right answer".
And, we can date things that have been dated by some other method.
In the 70's: If this sort of thing interests you, you should find the journal Radiocarbon and read one of the issues devoted to calibration.
For example, see Radiocarbon 46,1029 (2005), which has a calibration curve that goes back 26,000 years.
That causes a dating problem with any animal that eats seafood. After about ten half-lives, there's very little C14 left.
So, anything more than about 50,000 years old probably can't be dated at all.
Some were done by finding lakes with atmospherically derived carbon in their annual layers of silt (called varves).
In those particular lakes, the varves can be counted, and the varves can also be carbon dated. The first kind are datings of things that should't be carbon dated.
As the name suggests, fossil fuel is old, and no longer contains C14.