The smoke (and a loud screeching noise) was coming from the laundry room which is right next-door, so I ran in there and stopped the washer and the dryer, and smoke was coming out of the washing machine. It must've been a belt that broke - it sounded just like the vacuum does when its belt breaks.
So I called the landlord and talked to one of his vet techs (I've still never talked to him and we've been here almost a year!) and they called a repairman, who came in and, I assume, fixed it. I saw him come in and I told him what'd happened, but I didn't notice how long he was there.
And now today, I come home to a flooded laundry room. Someone else did their laundry, and the cycle finished, but either the sink overflowed or there's still something wrong with the washer. So I called the landlord again, and the same vet tech answered, and she told me to just call the repairman direct. (Of course, the landlord can't be bothered with his tenants' problems, he just owns the building!)
So now I'm waiting to hear. And in the meantime I'm going to go try to sweep up some of that water. I feel like an unpaid apartment manager! Whee!