Half life of carbon 14 is something like 560-580 years. After death it starts to become less. It can be applicable to all organic materials like bones, grains and woods.
For that reason it is widely used for archaeology. But is is not very secure to have carbon dating with something younge than 3-4 thousent years. And for things older than 50 thousent years, it is not also secure (because rest amount is so little). All carbon datings should be calibrated. The best is to calibrate with dendrochronology (tree rings).
The world is many much older than the age of 5766 years! It was just the times of Sumerians who invented writing (and of course reading! :-). Before the Sumerians many civilisations existed in Chalcolithic and Neolithic ages. Before that people were living with hunting and gathering. First stone tools of humans are dated to 2-2.5 milion years. Probably some people made tools with organic materials which cannot be left until these times.
At least for the humans, we can say a 4 million years of history. The earth is much older, like 4-5 Bilion (not milion!) years...