what size battery storage is needed to provide 10A at 115v for 5 continuous hours
From basic physics, (10 A)(115 V)(5 h) = 5750 Watt-hours.  That's what the system has to cough up.  The battery would have to store a bit more than that due to the converter from battery to 115 V AC having some loss.

There are many variables here, not the least of which are what daily insolation you expect the system to be able to recharge a empty battery with.  For example, let's say you decide to require 3 hours of full sun.  (5750 Wh)/(3 h) = 1.9 kW.  Let's say that together the converter from solar panel to battery, and the efficiency of the battery, and the efficiency of the inverter is 50%.  That means you need about 4 kW capability in full sun.



The answer to both these questions really comes down to "Do the math!".