What's the current draw of the air conditioner at 120V? If you assume 10 Amps, running for 8 hours, that's 80 Ah at 120V = 9600 Watt-hours. If you use something like 200 A-h Trojans, which are 6V batteries, they'll give 1200 Watt-hours each, so 8 of them will barely make it with a perfect inverter. Since you'll have losses, and you'll probably want some extra run time occasionally, and it's not good to run deep cycle batteries all the way down, you can probably figure 16 batteries. You'll need a good inverter, like maybe a 2 kW Exeltech, for example. Those are sine wave inverters and will do double their rating for surges, so it would have a shot at handling the starting current of the compressor.
A Trojan T105 weighs 62 lbs, so 16 of them add up to 992 lbs. At 120 bucks each, not counting shipping, that's $1920. The inverter is probably around $1,000. The charger will add some more cost.
LowTech said it right, a quiet generator is cheaper and a lot lighter. Maybe start with a very quiet one and put it in a sound enclosure. Some of the new motor home generators are amazingly quiet.