Japan has the oldest continuous, hereditary monarchy in the world -- and would, I believe, even qualify as oldest if we included non-hereditary or interrupted-hereditary successions. The Japanese royal family is still in the Yamato Dynasty, which took over the Japanese throne (according to legend) in about 660 BC under Emperor Jimmu. This means there has been a continuous line of descendants (traditionally, Japanese emperors must be male, but there have been several empresses in Japanese history -- though these generally succeeded in extraordinary circumstances) for nearly 2700 years.

However, evidence for the first two dozen or so emperors in the Yamato Dynasty is scanty by modern, empirical history standards, and the first emperors are semi-legendary in status. Therefore, the Japanese royal line is often dated back to the time when we have solid evidence in line with modern standards of historicity -- which takes us back to about 500 AD with the birth of Emperor Kinmei.

That said, even discounting that first 900+ years of semi-dubious imperial lineage, the Japanese monarchy would still be about 1500 years old, the longest continuous dynasty in the world by a significant margin. To give a historical context, the last emperor of the Western Roman Empire, Romulus Agustulus, was (probably) still alive about the time Emperor Kinmei was born, and Kinmei's dynasty is still on the throne today.

Sources:

• Packard, Jerrold: Sons of Heaven: A Portrait of the Japanese Monarchy (1987)

• Hoye, Timothy Japanese Politics: Fixed and Floating Worlds (1999)

Now I don't know if it is the longest, but the Chola dynasty had a clear, tracable family tree and record of succession from Vijayala Chola circa 840 AD to Rajendra III circa 1250 AD, that is 410 years of unbroken succession.

Now, the Chola dynasty itself is recorded in Sangam literature and mentioned in the edicts of Ashoka which itself dates back to 300 BCE, however we do not have any clear records of it is the same branch that went from 300 BCE to 1250 AD which would put it at a stellar 1,000 years.

The Pandyans, another major dynasty in South India trace their lineage back to the same Sangam era literature and have a recorded lineage / succession tree running from approx 300 BCE to ~ 350 AD (roughly 650 years). There is an interegnum and the second group of Pandyans run from ~ 500 AD to ~1250 AD and then the line switches briefly to a matrilineal one when the Pandyan empire was taken over by the son of Lilavati, a Sri Lankan Queen and then it runs continously till the Pandyan empire fell sometime in 1400 AD.

In essence, the Pandyan Dynasty per se was in recorded existence from ~ 500 BCE to 1,400 AD. They were contemporaries to Darius I and his Achmaemenid Empire and Rome when it was still a kingdom till the Pandyan dynasty / empire fell after the fall of Byzantium. The lineage though is broken up into 3 distinct lines, but their subjects and rivals as well as modern day historians consider the Pandyan dynasty as one entity.

Source - Cholas by Nilakanta Sastri and An Illustrated history of South India by the same author.

Follow-up: How about the current European monarchs? I know that Queen Elizabeth II goes back to the 10th century via Theodoric (since I'm related to her myself!), do any European monarchs go back further?