Which living person or family can believably trace their genealogy back the farthest? I understand that the further you go back, the more likely it is that a biological line ends, but so long as socially accepted family relationships are present, it counts (ie: a father having a son that is not biologically related to him by blood)
I believe that would be the family tree of Confucius, which has records dating back 2500 years. This is the main line of descent, based on the sources cited at the bottom of the page.
There are so many people who are related to this family tree today (last official genealogy had about two million members) that they have an official committee for maintaining the genealogy.
I was not able to find a link to the official document, but I have some records of an official document being published and presented (probably only in Chinese).
The Solomonic Dynasty can allegedly trace their routes back to the 10th century BC, though records become a little sketchy between 1000 BC and 1200 AD.
Although you generally don't see specific names, Jewish families do keep track of whether they're cohanim, which goes back pretty far, although how far depends on when you date the biblical period.
Could you clarify a bit?
The Japanese imperial family can trace their lineage back nearly 2,600 years with very good records. Confucius might be the better answer though since the Imperial line begins with the mythical "Son of Heaven."