Can a capacitor provide enough current to be used as a fail safe for a servo?
Yes, you can do this with supercaps.  According to this calculator, 400,000 µF or 0.4F will give an RC time constant of two seconds with 5v and a 5 ohm load (i.e. 1A).  Half of that (1 second) will drop the voltage of the supercap to about 60% of its voltage, or 3v.



To make up for the constantly falling voltage during discharge, you can connect the supercap up to a boost converter like the ISL97519A, configured to have a 5v output.  You can "or" the output of the boost converter and your regular supply together with two Schottky diodes.  The boost converter has an enable lead that you should normally assert low, and then when power is lost, keep it high by tying it to the output of the cap.

Most supercaps in the 1F range have a voltage rating of 5v or less.  To get a higher rating, you can put two in series, although this halves the capacitance.  The Eaton Bussman PM-5R0H155-R is rated at 1F, 5V with an ESR at 100 mOhm.  This ESR is higher than we'd like, especially since we have to double it so the voltage drop will be 200 mV.  However using a boost regulator will mitigate that.

The boost regulator and supercap are available from Digi-Key for $3.50 and $7.54 respectively.