How to store and step 12V to 5V with capacitor and regulator?
Capacitor C1 discharges very quickly, why is this?


Quiescent current (no load current) for a 7805 is about 5mA. 

Q = CV or... 

\$\dfrac{dQ}{dt}\$ = C \$\dfrac{dV}{dt}\$ = current

So 5 mA *\$\dfrac{1}{C}\$ = \$\dfrac{dV}{dt}\$ = 106 volts per second.

Or put another way, if you draw 5mA from a 47 uF capacitor charged to 12V, expect it to be fully depleted after 113 ms.

If instead of the 7805 you chose an LM2936 this time period would extend: -



Iq for this device is less than 15 uA for a no-load situation. This will extend the time period by about 333 times.