How does the Doctor go to Gallifrey in "Day of the Doctor"?

"It has been said earlier that the TARDIS cannot travel to other universes", you say. But then again, the TARDIS has traveled to other universes. Mostly in the very same episodes where it was said to be impossible. It traveled to the parallel universe in Rise of the Cybermen/Age of Steel in series 2, and to the pocket universe where House lived in The Doctor's Wife in series 6.

What this tells is that that first, everything is wibbly-wobbly in Doctor Who, and rules are established only to be broken, hopefully in a dramatic and satisfying way. And also, to keep to in-universe explanations for a second, that while normally the TARDIS can't travel between universes, special circumstances - such as Torchwood's use of the Dalek void ship to link the universes in Army of Ghosts - can certainly create situations where it's possible. The Time Lord technology used in the painting is that special situation.

If I remember correctly, it was said that she is not supposed to travel to other universes. In the times that it does, generally bad things happen like she starts to have power failure, etc. It does not mean she can't do it, just that it isn't necessarily "healthy" for her.

And besides, timey wimey wibbly wobbly.

They didn't fly the Tardis in.

Galifray is timelocked, they can't get in. They can however enter the painting in the same way the Zygons did.

The Doctors can't fly into the Tower of London because it's been "Tardis proofed". Instead they walk into the painting. What is a little unclear is whether they can travel back into the painting, I suspect so as they are not visible in the early scene when they examine the picture (with the broken glass).

• OPTIONAL they travel back in time before the painting is examined - this is unclear whether you can travel back in a painting
• They call back/forwards in time to the museum instructing them to move the painting into the secure office
• They enter the painting and wait
• They exit the painting to enter the Tower.

Regardless of whether they entered the painting before or after it's moved to the tower is largely unimportant, however the key point is that they didn't fly the Tardis into through the timelock, they walked into the painting in the same way the Zygons did.