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Carbon-14 is a radioactive isotope formed in the upper atmosphere.
Thus even from a 'young-Earth' standpoint, all radiocarbon dates (assuming that care is taken to eliminate contamination) are taken to be meaningful indicators of a given sample's age. Anyone familiar with typical studies employing the radiocarbon method knows that model ages obtained often exceed 10,000 years (e.g. So doesn't the method already affirm that the Earth (or at least it's now deceased inhabitants) must be at least this old?
Plants absorb that carbon through photosynthesis, and animals via plants.
When they die, however, the radioactive carbon decays at a known rate.
Theoretically, radiocarbon techniques have the ability to date samples to around 75,000 years, but the working threshold of reliable dating is around 50,000 years.
Samples significantly older than this have very little or even no measurable C left.