I have a few ideas that haven't been mentioned yet.
An ice hockey rink (by NHL rules, which we'll use since it's smaller than international rules) is 200 feet long and 85 feet wide, and has rounded corners with a 28 foot radius. This gives a floor area of about 16327 square feet, or 1517 m$^2$. It doesn't need to be very tall, so we can just give it a comfortable ceiling of 3 meters, for a volume of 4551 m3. Hockey is played with six people on the rink per team (penalties notwithstanding), so there are 12 people active at a time. This gives a space efficiency of about 380m3/player.
The banked tracks for bicycle racing (velodromes) are specially built, with no strict dimension requirements that I could find, so I will use the parameters that I could find to make some estimates. By the rules used by the Olympics, the track is 250 meters long by a measurement near the inner edge. It is an oval, with two straights and two 180-degree curves. While the exact dimensions are not standardized, one measurement I found gave a 16.6 m radius for the curves, which gives straights that are about 73m long. Most tracks are 7 or 7.5 meters wide, and there is also a 4 meter wide safety zone on the inside of the track proper. Using these dimensions gives a surface area of about 3150 m2. (This does not include the center of track, which could be used for other purposes, though that would interfere with the spectator value.) A 3 meter height seems reasonable here, too, so that gives a volume of 9450 m3. This estimate is probably a little high because of the banked curves, but it's close enough for government work.
One advantage of a cycling track (or any type of track) is the variety of events it can host. There are two-person races or even one-person time trials, and both sprints and endurance races. This means that a track could be in use more often, making the space more time-efficient. This variety also includes the points race, which is a mass-start race involving a large number of riders. If we allow for 30 riders (the Olympic event has had as many as 28 participants, so this seems reasonable), then we get a potential space efficiency of 315 m3/rider. There are also "Madison" races, which are like tag-team racing. Again going by the Olympics, this can have as many as 36 riders, in teams of two. Since half of the racers are "resting" (but still riding) at any time, you might not allow this count. If you do, that improves the space usage to 262.5 m3/rider.
One other consideration is that reduced gravity could be to your benefit, allowing you to reduce the volume used by increasing the banking angle of track. Or, if closer to Earth-normal gravity, the inner safety zone could double as a running track when the bicycle track is not in use, giving more use to the same space.