I believe that the Yamato dynasty of Japan is the longest reigning royal family, having reigned since 660 BC according to legend, and at least since the early 6th century verifiably -- but they are probably so long-reigning precisely because they became figureheads fairly early on, so I'm not sure they meet your criterion for "they stayed in power." A discontented daimyo wanted to overthrow the shogun, not the emperor. (A few periods of unusually influential emperors, such as the 1868-1945 period, notwithstanding -- although even in that period, the real influence of the emperors is a matter of controversy.)

There is also a (not sure how debatable) long line of rulers of present day Ethiopia that supposedly dates back to Axum and pretty much ended with Haile Selassie.

EDIT: here's a link to the line of emperors going back to the 1100s and another of the kings of Axum going back to the 100s.

Wikipedia says the links between the two are not solid.

The official story in Ethiopia I think is that the ruler is descended from the Queen of Sheba's (and Solomon's) son, Menelik I, claimed to be the first Emperor of Ethiopia.

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I would think the Hapsburg dynasty is pretty high up there as far as European history goes. Something like a 300 year reign.