Good Batteries for (way) Sub-Zero Temperatures
I agree with the previous answer: Five volts at 25 mA is a lot of power to fit into that small a volume, unless you need it only for a short time. I just grabbed the spec for a standard cylindrical Lithium Ion battery, and see that it has a capacity of 2400 milliAmp-hours, in on order of the total volume you have available. If you had nothing but that volume of battery and a load resistor, you'd get just under 100 hours at 25 mA, though at only the nominal battery voltage rather than 5 Volts.

If you have a short "mission time," where your device sleeps until apogee, then draws 25 mA, you might get away with something like a super capacitor. You'd need a switching supply to convert the (dropping) output voltage to the constant 5V you want.

A better answer approach might be to re-examine your power requirements. Could you use a smaller processor, one that might run directly off a small battery, at much lower current? (You can slow down the clock on many parts to reduce current draw.)

The PongSat page to which you provided a link shows photos of two devices with what look like standard coin cells. While that approach might not produce a design that could go into volume production and reliably work at low temperatures, it might suffice for a one-off unit.