Hmm....haven't done this before, but sure I'll play for a bit.
I guess you should start with how many watts you need for the lights.
4 of those LED shop lights? They should be around 38~42 Watts each, so lets say 170 Watts and drag it up to 200W for a nice round number and some other loses.
Regular 120V household AC voltage for the lights means you need about 1.7 Amps if we round up. A bit more amps if you run 115V and a bit less if you run 125V.
Now you want them to be on 4 hours or more at a time? So how long can a deep cycle battery put out 1.7 amps.....(math is easier with 2 amps

)
so 4 hours times 2 amps equals 8 AmpHours....which you might be able to use a small battery, like a motorcycle battery, but that's 4 hours and total capacity, maybe something bigger since I haven't factored in any losses from the 12V DC to 120VAC converter. Should I even be using 12V? Or something more like 24V?
I'll have to think about this some.