EE questions

4 times 120 watts = 500 watts
plus inverter efficiency. probably 550 to 600W out of he battery.
600 w/ 12.6 volts (asuming a 12 volt inverter) = 47.6 amps
then why bother? the batteries will be the biggest part of the cost.
lets assume you use 100 A-H deep cycle batteries
you would think 5 would give you 500 A-h for 50 amps times 10 hours but in reality you will need more.
you would, depending on the battery type, charge each battery with its own charger @ 10 amps for at least 10 hours (more likely 15 hours).
i'd pick a 1 kW or larger. make sure it will operate with lower voltages. some (like one i bought) are intended to run in vehicles under charge (i.e. the engine running) once the voltage drops to 11 it shuts itself off. i guess the idea here is to not kill the battery and leave you a chance to get the car/truck started.
you could also opt for 4 smaller systems, say four 200 W inverters (or UPS units with the battery cables brought out) each going to a couple of batteries. keep in mind that the charging circuits of small UPS will not be able to cope if these are to be used daily.

I really don't think you can build cheaper than you can buy, but if I had to I'd use car batteries wired in parallel along with a regular inverter intended to be conencted to a car battery. I'm not sure exactly what I'd use for a charger, but presumably a 10A 12V charger would work. And all of this could be stolen just them same, but you know what they always say about the boss being right!