Who's Doctor Who? Do you mean the Doctor? Because the Doctor's about 1300 years old.
Doctor Who, the TV show, was first aired on 23 November 1963. You can work out its age from that but the result will change as time passes.
- Assuming the asker is referring to the Doctor, the fact is - according to Steven Moffat - no one knows his age, not even the Doctor. In the original series, the last age given by the Doctor on screen is 953 at the start of his seventh life; the novels pushed his age well beyond 1,000. When the series returned in 2005 he was claiming an age of 900. (As River Song says: "Rule No.1 - the Doctor lies.") Both Russell T Davies and Steven Moffat have talked about including a scene in which the Doctor says he no longer remembers his age, but it's been cut for time at least once, possibly twice (i.e. It's not considered that important). Due to the events described in "The Impossible Astronaut" and the lead-up to "The Wedding of River Song", the Doctor was said to be more than 1,100 years old. The recent episode "A Town Called Mercy" has established that he is now claiming an age of more than 1,200.