How to calculate capacitor required?
Capacitance in Farads F = Current I x time t / voltage dV

If the load current is limited by resistors and not a constant current load, the current will drop as voltage reduces, so the above calculation needs to be changed to a I x dT / dV calculus form where dT tends to zero. This becomes a bit more messy. So, let's keep things simple:

For the output case with a 3 Volt drop over 10 seconds, assuming constant current load, capacitance works out to:

F = 0.03 x 10 / 3 = 0.1 F = 100,000 microFarads.



Such "super capacitors" or "ultra capacitors" are available online fairly inexpensively these days, if the purpose is for a DIY project. For instance, on eBay, 3 pieces would cost $4.50 from this seller (plus shipping), or a single piece for $1 from a Canadian seller, if that saves you some shipping cost:

 

That is the capacitor needed. 

The charging period and supply voltage have no effect on the capacitance calculation, assuming your supply can provide the required current (300mA), so ignore those.