What Kind of Battery Can Power This Fan?

You were told wrong.

It's done all the time.

i dont know why i posted that as anonymous, opps.

ANYWAYS, then why cant a battery power a peltier chip for any length of time?

http://www.amazon.com/s?ie=UTF8&rh=i%3Aaps%2Ck%3APeltier%20Chip&page=1

But they can.

The key always is how long any load must be driven by a finite size battery. Depending on what you are trying to cool and how the transferred heat gets removed, you may be able to lengthen operating time with control circuitry so that the cooler does not have to be ON all of the time.

You need to compute your load in terms of amp-hours times voltage, and then compare it with possible batteries.

These motors are "DC brushless" and cannot be connected directly to a battery hoping they run, an electronic driver circuit called an "ESC" must be used.

Batteries used by the R/C enthusiasts in this applications are from the old NiCd passing thru NiMH and ending in the LiPo.

Funnily in that same webstore you'll find all the parts needed to run that motor.

Bear in mind that autonomy of an electric R/C plane is in the range of minutes of flight.

If your requirements are greater than that, you'll need to take into account:

heat dissipation for the ESC (yes they get VERY hot) and battery capacity.

That fan sounds like what it is... a 30,000 rpm monster. It makes so much noise that you couldn't stand it. Also, it's useful life is measured in hours. It can't be built into anything permanent. The illustration shows three leads. An ESC must be inserted between the battery or source of high current DC and the motor. All in all, it is NOT what you are looking for here.