They are, obviously, assuming the amount of carbon-14 in the atmosphere has always been constant, and its rate of decay has always been constant.
Neither of these assumptions is provable or reasonable.
An illustration may help: Imagine you found a candle burning in a room, and you wanted to determine how long it was burning before you found it.
You could measure the present height of the candle (say, seven inches) and the rate of burn (say, an inch per hour).
In order to find the length of time since the candle was lit we would be forced to make some assumptions.
We would, obviously, have to assume that the candle has always burned at the same rate, and assumes an initial height of the candle. Similarly, scientists do not know that the carbon-14 decay rate has been constant.
When a plant or animal dies it stops taking in air and food so it should not be able to get any new C-14.
This would prove the earth is not yet 30,000 years old!
This also means that plants and animals that lived in the past had less C-14 in them than do plants and animals today.
At some point you would be putting it in and it would be leaking out at the same rate.
You will not be able to fill the barrel past this point of equilibrium.