So I bought 4 Multistar Elite 3508-268kv electric motors and 4 Afro HV escs. I burned out one of the motors somehow. For all I know, it could have been a defective unit when I bought it because I never saw it running. I am also pretty sure I burned out the esc.
What happened was that when I plugged the motors and escs in, 3 motors were working and one wasn't. So I calibrated them all manually. Still 3 worked and 1 didn't.

One thing I tried was connecting the not-spinning motor to a different esc. It still wasn't spinning. I also tried connecting a functioning motor to the possibly-not-working esc. The motor didn't spin however when I tested the esc's leads with a voltmeter, it was making voltage when I had the throttle turned up. (weird)

After messing with it all day, the esc that the not-working motor was attached to started blinking a red light instead of a solid green one. I then updated that escs firmware (using the correct firmware and following all the instructions in the manual) and went through the calibration again. When I throttled up to see if it was working, the motor started making a buzzing noise, slowly spun for almost a whole rotation and then the esc made smoke so I immediately unplugged it. The esc still turns on (but with a red light) so it's probably dead.

The reason I'm sure the motor doesn't work is because when I connect a voltmeter to the leads and then spin it with my finger, I get no reading. When I do the same thing on one of the motors that I have seen working, the needle flickers.

Is there anything I did wrong?

It doesn't sound like you did anything wrong. When you start up a motor the first time and it doesn't work, it's a defective motor. It's possible you had an internal short or something binding inside the motor which would have caused a big current draw that burned up the ESC. I'd take a shot at getting warranty service from the vendor on these since they didn't work at the initial start up and the other motors did.