Batts / inverter / window AC unit.

You didn't mention the duty cycle... most air conditioning units cycle the compressor on and off. 

Anyway, your load is 5 amps X 120 volts = 600 watts.  Divide by 0.85 inverter efficiency = 706 watts from the battery.  

706 watts ÷ 24 volts = 29.4 amps.  Your battery is 244 amphours at 24 volts.   That amphour rating is based on a amp draw of 244 amphours ÷ 20 hours = 12.2 amps.   Your draw of 29.4 amps will have the effect of reducing the capacity of the battery (Peukert factor). 

Another factor working against you is the modified sine wave.  That will not be good for the air conditioner and may make the system less efficient, reducing the run time on the battery. 

You didn't mention how far down you intend to discharge the battery, so I can't answer your questions without knowing that (as well as the duty cycle).   If you draw 29.4 amps for an hour, that is 29.4 amphours of energy.  29.4 amphours ÷ 244 amphours = 12%.  Thus your battery would be drawn down to 88% SOC (state of charge)...  except that it would be lower than that because of Peukert effect.

One other factor worth mentioning... the current (amps) you draw from the battery increases as battery voltage decreases.  Thus you are drawing more amphours every hour from the battery as it discharges.

--vtMaps

Lots of problems even doing a seat-of-your-pants estimate, which I guess you've seen.

I have used a 5200 BTU 10.7 efficiency Kenmore (no reason to be poor efficiency ACs) on 4 - 6 volt 215 amp golf cart batteries, in a 24 volt 215 amp battery bank.

Cabin was built in the shade with 6" thick insulated walls. 10 x 16 x 13 tall. Fully charged they would runabout 80% duty cycle for the first hour and roughly 40-50% duty cycle average for the next 5-6 hours. 450-475 watts (at @115 volts) run from a true sine 1800 watt inverter(Prosine). I figured roughly 50% of capacity used each night, batteries lasted 4 summers of use like this (they were a year old when torture began!) Though I did add to my array so I could cool during the day as well as fully charging the battery bank, so the early duty cycle was lower since the cabin was starting at a cooler temp.