There is some evidence of it going farther back to 9,000 years, essentially trying to harvest wild bee honey without killing the bees themselves, however 5,000 years is generally considered the time period when honey bees first became domesticated and we were actively managing and colonizing multiple artificial hives.
I mean it was probably easier 250,000 years ago for primates to steal some honey, because we still had our long body hair back then I think, so that's more protection from bee stings.