So I went upstairs to the on-site manager, and she called the prompt service person, who has come and shut down the hot water heater (which was in said closet). Now she (a very nice butch) and I are shooting the breeze waiting for the night plumber to show up.
(Update. The water heater company finally called. They will come out tomorrow, sometime. HOPEFULLY they will call be before they come. Sigh.)
The on-site manager also let me know (unofficially) that this row of buildings is going to be demolished at the end of the year, to be replaced with 18 800K townhomes. (So that's why they didn't ask for a lease renewal when the lease term expired!) Official notice will go out 60 days prior, but it's time to start thinking about a new living situation.