Cloisters

The Cloisters were an area located deep beneath the Capitol on Gallifrey. The Cloisters housed the Matrix, where the minds of dead Time Lords were stored, and a set of Cloister Bells that rang in the case of impending catastrophe. They were defended by Cloister Wraiths and by alien invaders that had been neutralised and "filed" by the Matrix; these forces prevented anyone from leaving the Cloisters once they entered. The Cloisters were connected to the old workshops where TARDISes were built.

History
Early in their history, the Great Houses found that the construction of new timeships was so complex, involving the participation of the future timeship itself, they could only happen in zones where the normal laws of time were suspended. Thus, a special "null-zone" connected to the caldera was constructed. It was buit to be "connected" to the crater while "causing as little disruption to the area as possible". Its machinery also allowed information to be drawn from across time and space, and it came to be referred to as "a kind of 'womb'": (PROSE: The Book of the War) the Matrix. The old workshops where TARDISes were constructed were directly connected to the Matrix's Cloisters. (TV: Hell Bent)

When the young Doctor was at the Time Lord Academy, he entered the Cloisters and was shown a secret passage out by the Cloister Wraiths. The Time Lords also believed that they had told the boy the prophecy of the Hybrid and told him what the Hybrid was, potentially himself, which may have been one of his motives for later running away from Gallifrey. However, he refused to confirm this story when Me recounted it to him, and implied to Clara Oswald that he did not know for certain what the Hybrid was and had "convinced the Time Lords that [he] knew" for his own purposes.

At any rate, (TV: Hell Bent) at some point in the Doctor and the Master's lifetimes, the Cloisters were the site of the Cloister Wars.

During the Last Great Time War, President Rassilon had a TARDIS carrying Romana and Leela diverted to the Cloisters so he could confront them privately. There he had Romana banished to a pocket dimension and conscripted Leela. (AUDIO: Homecoming)

After escaping his confession dial onto Gallifrey, the Twelfth Doctor took Clara Oswald into the Cloisters to prevent the Time Lords from returning her to the moment before her death. The Doctor then stole a TARDIS and took him and Clara towards the end of the universe. There, in the now-ruined Cloisters, he found Me waiting for him in a bubble of reality. (TV: Hell Bent)

In an alternate timeline, the Doctor had a romantic relationship with River Song since his first incarnation and would smuggle her into the Cloisters. (AUDIO: Someone I Once Knew)