This is a gymnasium in the grand old sense: It's got baths, steam rooms, massage tables, an exercise field, pools, lounges, and even a portico where the teachers of the Transcendent Order instruct their students. All of this is enclosed in a great compound of gold-veined black marble. The Gymnasium is open to all, but only on the Ciphers' terms.
Compared to other parts of Sigil, life here is deliberately unhurried. The Transcendent Order (whose faction headquarters these are) believes understanding can only come with a calm mind, so they do their best to keep the pressures and concerns of the outside world at bay. Those who enter must leave weapons and magic behind. No spells can be cast there, nor are beings with innate powers allowed to exercise their talents. 'Course, none of these rules apply to the Ciphers, although these edicts are generally followed by most of them, too. Nor are the rules perfectly obeyed by visitors. There are always little incidents to disturb the perfect calm of the place, disturbances the Ciphers have to put down.
Because of the rules and services here, the Gymnasium serves two purposes. First, it's a place for citizens to relax and forget the cares of the world. Noise, pressure, even social class can be forgotten. Second, the Great Gymnasium serves as a neutral ground for hostile parties. Many a truce, treaty, and pact have been negotiated in the steam rooms and baths. Like every other part of Sigil, the Great Gymnasium is vital to the functioning of the city. If it didn't exist already, it'd have to be created.
The streets around the Great Gymnasium are host to a score of smaller competitors, so the whole district is noted for its baths and spas. Some are general while others cater to particular races or planes. With the baths come inns and food shops, many of which offer nothing put healthful exotica.
Note: The Great Gymnasium of Sigil originally appeared in the Planescape AD&D campaign setting from TSR. Additional information can be found in In the Cage: A Guide to Sigil. Their use here is for the purposes of providing context for the campaign only.