The game of cricket has been mentioned as an activity since at least 1598, when the first reference to the game was recorded, discussing an incident of the game being played in 1550. While individual sports like wrestling and athletics date back to the ancient Greek Olympics, cricket is probably the oldest modern team sport that is still widely played.

Source:

History of Cricket Volume 1, by H. S. Altham

Golf is a current popular sport that is dated to be between 400-700 years old. There are quite a few peices of text from that time all pointing to what appears to be the beginnings of our current day golf.

I would imagine any traditional track sports could be considered as well - going back thousands of years. I would personally throw in equine sports (although they can be taken as more of an event than a sport.)

Robin Bargmann, Serendipity of Early Golf, 2010

The sports in the Ancient Olympic Games in Greece were:

Boxing, Chariot Racing, Horse Racing, Pankration, Discus Throw, Javelin Throw, Long jump, Running, Wrestling,

All of these except Chariot Racing and the Pankration, still exist, and are still popular. The Long jump has changed, as Ancient Greek athletes jumped carrying weights which they threw backwards as they landed. The rest are still today much as they were in 776 BC, which is over 2700 years ago.

All these sports have remained popular for a very long period of time.

Golf has a history going back to 1297. Though that exact date might be apochrophal.