Baseball is a game of longevity. In contrast to basketball or football in particular, baseball players can play well into their 30s and have a great deal of success. Every now and then, you even get a guy like Jamie Moyer or Julio Franco who makes it almost to 50.
Interestingly, there are many free agents who will be in this position this winter. They may be at least 35, but they seem to have plenty of gas left in the tank if 2013 was any indication.
(Note: ages listed are for the upcoming season.)