If we look at this as skeptically as possible, accepting only the most well-attested historical records, the Imperial House of Japan is the best bet. The Solomonic dyansty in Ethiopia claims to have ruled earlier, but a lot of their claims are very poorly documented and they don't even claim to have ruled continuously.

Beyond that, there are a few other houses of note. I'd recommend looking at the Rurik Dynasty, of Russia, and the descendants of Hugh Capet. While the formal Capetian Dynasty didn't hold the Kingdom of France for all that long in the grand scheme of things, descendants of Hugh Capet held power for a ridiculously long time through cadet branches like the Bourbons, and are still on the thrones of Spain and Luxembourg today.