Track itself only has discrete roots back to the early 1800's in various countries in Europe, eventually making its way into America. Track events however, started back before there were even countries that one could classify. The Greek started holding Olympic Games including events such as the Standion Race (Running Race), the Long Jump, the Javelin Throw, the Discus throw, and wrestling. All of these (with the exception of Wrestling) are modern day track meet events. So to directly address your question: Track and field began nearly 2700 years ago in the Greek Empire.
Track and Field