If you mean "how old was Jon Pertwee when he became the Third Doctor?", then Jon Pertwee was born on 7th July 1919 in Chelsea, London. He was cast as the third Doctor in the summer of 1969, around the time he reached his 50th birthday.

If you're referring to the Doctor himself (the character, not the actor playing him) no explicit age is referred to on-screen. In Tomb of the Cybermen the Second Doctor claims he is 450 years old. In "Pyramids of Mars", the Fourth Doctor states that he is 749 years old.

Logically, the Third Doctor should be older than the Second, but younger than the Fourth. Say, approx 600-700 years old (give or take 50 years)

Logically, yes. However the 7th was claiming to be well over 900 long, long before the 9th claimed to be 900. Various people connected with the show have said, on numerous occasions, that we can't take the Doctor's word for his age because he long ago lost track of it. He just doesn't know how old he is. It's all that time travel, you see.

The 3rd Doctor also implied that he was thousands of years old in "The Mind of Evil", deepening the confusion.

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