Battery size calculation
A 40uF capacitor charged to 300v requires \$0.5CV^2\$ = 1.8Joules of energy. Once per second, that's 1.8W, assuming no losses. 

Charging a high voltage capacitor is best done from a variable voltage step-up current source, for which a flyback boost converter is a perfect fit. This will obviously have some losses, so we'll need more than 1.8 watts.

If you had a 1000mAh LiPo battery, at a nominal 3.7v, that would store 3600*3.7 = 13320 Joules. This would power a 1.8 watt load for 13320/1.8 = 7400s = about 2 hours. Allowing for losses, you might hope for more than one hour.

If the battery had a different voltage, then the energy capacity sums would be different. With a different, perhaps lossier method of charging the capacitor, the loss estimates would be different.