The 4th Doctor famously said, 'Sleep is for tortoises', and the 11th Doctor apparently went off and had adventures on his own and with River while Amy and Rory slept, but we do know that he sleeps at least occasionally, especially after a hard regeneration (4-5, 9-10), and we know he had a cot when he was a baby ("A Good Man Goes to War"). So it's likely that he has a bedroom.
This rarely comes up on screen, and when it does, it's never answered. At one point, Rory wanted to ask about the Doctor's room, but Amy didn't let him. In the DVD mini-episodes for series 6, Amy asked if he even has a bedroom, but she got distracted and asked a different question before he could answer.
In the novels, he definitely does have a bedroom in the TARDIS (although it's implied that the 1st Doctor did all his sleeping on the lounge in the console room), as well as one in his house in Kent, and had one at his house/condo/whatever on Gallifrey. It's left pretty open how much time he spends there, but at least some incarnations have said they sleep at least an hour every night, or that they enjoy sleeping.