Summary: The Doctor dislikes being thought of as old.
Just because the Doctor looks like an old man, with his lined face and white hair, doesn’t mean he is one. Well, he is and he isn’t, he’s certainly old enough to qualify, but he’s a Time Lord so the same rules don’t entirely apply to him. He could live for centuries and wind up looking younger than his granddaughter’s teachers.
Age is just a number, and appearances can be deceptive. The Doctor is as spry now as he’s ever been, not the doddering old fool these young whippersnappers take him for. Give him a chance and he’ll prove it.