Can I use a single AAA battery to power a motor?
Yes you can but there are efficiency issues so it's not exactly easy to determine the runtime.
You need to use a step up (boost) power converter. It will take your 1.5V and with some electromagnetic work transform it in 3.1V or whatever (there are limitation but this is feasible).
Now, the available power doesn't change (to be picky we are talking about energy but it scales), so if you have 700mAh at 1.5V and boost it up to 3.1V (to simplify the calculation) you'll have about 340mAh available.
Also the converter to work need to use some of that power. A good starting point is 80% of efficiency (depends on many thing). So you are left with 0.27Ah available for your motor.
Ignoring other losses (motor controlles and other circuit parts) you need to determine your actual load: a stepper without microstepping always drives one or two phases, depending on the drive mode. Assuming half-stepping in average you are driving 1.5 phases, so you sink 1.8A; with 0.27Ah this should be about 9 minutes (unless I flunked some calculation)
There's another big thing to handle: your battery is 700mAh but the effective capacity depends on the current actually sinked from the battery (it depends on the battery technology). So even if theorically it should handle 700mA for 1 hour or 350mA for 2 hours, it's quite probable that in the second case it will last longer.
In the end, it depends on how the 700mAh rating was determined, if with a constant sink or some standard power profile: many consumer batteries are rated with the 'digital camera' profile, with a relatively low constant load and only some high current pulses.
So, it's difficult to answer, you'll need some testing. I'd recommend to acquire an evaluation board for your booster of choice and work from that.