An extraction chamber was a place where a person was extracted from the last moment of their time stream for questioning, time looping their physical processes and leaving them frozen between one heartbeat and the next while keeping them conscious and aware. When the questioning was done, the extracted individual was returned to their last moment without damaging the timeline. At least seven existed in Gallifrey's Capitol.
Clara Oswald was taken to extraction chamber 7 at the point of her death supposedly for questioning about the Hybrid. This was, in fact, an act of desperation on the Twelfth Doctor's behalf to save her life. Because Clara's death was a fixed point in time, she intended to return to Gallifrey, "the long way round". (TV: Hell Bent)