After having to shower with cold water for 4 days, the water heater has been fixed. It turns out that it was improperly installed. The water heater in my garage, as I learned last night, is slightly different than most. It's a direct-vent model.
I don't know much about these things, but basically, there's an intake tube and an exhaust pipe that sits inside it. And, for whatever reason, the main outer tube was sealed tight as it should be, but the inner tube was not sealed at all. The end result was the burner would not have enough oxygen to consume, which in turn would lead to the flame going out completely.
The water heater guy was here for over two hours diagnosing the problem and re-installing the venting part of the system, and not long after I had hot water again. One annoying thing though... I'm going to have to get reimbursed for the $239 repair. The builder and its contractors thought it was a defective unit, and they contacted the manufacturer, who sent out the guy that came yesterday. But, since it was not a warranty issue, I had to cough up some cash to pay for the repair, since it was already late when the guy showed.