If you want to start watching the modern Doctor Who, then you've got 8 seasons to catch up on. That comes out to (I think) 122 episodes. That's a lot, but it's feasible. It'll just take a few months.
Now classic Doctor Who--the show than ran from 1963 to 1989--that's a different story. Then you've got 707 episodes. To the extent that watching TV can be considered an act of work, it's a huge chore.
But don't worry. This video will help you get caught up very quickly. You'll just watch all of the episodes simultaneously. It's helpfully annotated with the names of the actors who played the Doctor.
-via David Thompson