The Crab Nebula was a supernova remnant in the Beta Quadrant. This nebula was visible from Earth in the constellation Taurus. It was approximately 6,500 light years from Sol.
In 2365, attempting an amusing anecdote with the aim of entertaining Captain Picard and Lwaxana Troi, Lieutenant Commander Data explained how he was able to precisely calculate the distance between the Crab Nebula and the Omicron system. He accomplished this by multiplying the square root of pi with nine to the third power. (TNG: "Manhunt")
This nebula was created by a supernova that was witnessed on Earth in 1054 AD. It contained the first pulsar to be visually observed by astronomers and was a strong source of high-energy radiation.
The nebula's quadrant of origin was inferred from the location of other known astronimcal objects which were associated with the constellation Taurus.