It doesn't actually require extra tracks to run a 24/7 metro and also do maintenance. A very common solution is single-tracking. BART has two tracks, one for trains in each direction. What some systems do is just close one side for maintenance, and then run trains on the other side in both directions. For low-frequency service (e.g. hourly night service) it can just be a single train shuttling back and forth end-to-end. For higher-frequency service a few other options are possible. You can split the route into smaller segments (say, in half) and have one train shuttling back-and-forth on each segment. Customers change trains at segment boundaries if they want to go further. Alternately, some systems can run trains fully in opposite directions, if there is a siding somewhere along the route that can be used for passing.

The Copenhagen metro is a 24/7 system, without extra tracks, that uses this method. On maintenance nights a usual pattern is a brief complete-service shutdown 1:00-1:30am to switch from regular operation to single-tracking. One track is taken out of service and powered down, and service at ~20-minute headways resumes on the remaining track. The Atlanta subway also uses this method for longer weekend maintenance sometimes, going to single-track mode all weekend (they don't do 24/7 operation, though, I would guess due to budget issues).

It's possible BART can't do maintenance with single-track operation due to how it's set up, but their message implies it's physically impossible without having spare tracks, which is not true in other systems.