The Doctor has the Tardis but that could be compared to a caravan in a sense, a moving home, where you have no base.
Does the Doctor want somewhere to call home in the proper sense. Is that why he travels everywhere, is that what he is really looking for. Does he want to go home?

No, he doesn't.

The Doctor occasionally referred to himself as an exile in the early years, but from about the time of the second Doctor onwards it's fairly clear that he is a traveler at heart and has no desire to go home. In various spin-off media over the years, it's established that he does have residences on Earth and elsewhere (like the house in Allen Road in the New Adventures novels) but these always seem to be retreats or holiday homes rather than an actual home, and are rarely visited. I think the only times he occasionally exhibits a bit of regret about leaving Gallifrey are at times when, as in the first few years and now since the Time War, he actually can't go home. It isn't that he particularly wants to. He just likes to have the choice.

Hell no! He's never leaving sexy! Not until the day he dies!

the doctor also stated in the impossible planet that he would hate having carpets and doors and all that crap also mortgage.

I think that the Doctor has his job to do. He has companions. If in danger he clicks his finger. Enter the little hamster.

Please! This place is twisted enough at the best of times without you suggesting that the Doctor lets off steam by entering little hamsters!

Calm down ting. Check out sandydunes pet thread. Sandy and I have been script writeing.!

I'm not sure if hamsters are canon...

nyingy

Probably not canon, but ever read the comic strip "Fellow Travels"? It reveals The Doctor has a house in the country he likes to settle into over the years in between traveling. He and Ace returned to it in the story.

not even to visit for the Christmas hols?

Sexy could come with.

the trusty friend.

That's the same house seen in the New Adventures novels, I think; the house in Allen Road, which I mentioned earlier. At that particular point, Marvel were deliberately tying the comic strip continuity into that of the book range, something the comics had never previously done (and would later abandon). Most of the earlier and later strips are clearly off in their own parallel universe somewhere, but those few seventh Doctor strips may actually be canon, depending on your point of view.

Yeah, I have to cherry pick which strips are and arent canon, I know "Ground Zero" was decanonised by "Death of the Doctor" referencing "Dorothy" running A Charitible Earth, indicating she's alive and well.

Although since "time can be rewritten" that probably ticks all boxes.