I use a 600W pure sine wave inverter connected directly to my car battery, or a separate 26ah gel cell. I can run two 800ws strobes off it quite easily, and have run two 1600ws strobes without any issues. I also use a Honda 2000EUi generator (about 18amps) that has a pure sine waver inverter internally, for when I want to run 4 lights. Basically, the generator gives me a single 20amp household circuit which I can easily run 4-6 heads without any issues. I've only tested the generator w/ three 800ws and one 1600ws so far and it worked perfectly.
When using a single 1600ws or two 800ws heads on inverter and battery setup, I've usually gotten about 400 or so flashes from the battery. I have a second battery that I can swap out if I exceed the 400 flashes. If I run at reduced power levels, I can get far more flashes per charge on the batteries, but haven't tested the number of flashes per charge.
I use the same inverter as the Alien Bee Vagabond setup, but use a larger battery. You can use the table on their site to get an idea of the number of flashes based on the power settings, and then recalculate that to include the extra power of the 26ah battery over the one they use (15ah, 20ah, or ??).
So yea, in short, no problem. Just make sure to use a pure sine wave inverter, and I would recommend 300w capacity per head you plan to use on a regular basis. Ie, 2 heads, 600ws, 4 heads, 1000w or 1200w intverter, etc). It will reduce the load on the inverter and provide faster recycling time.
Keep the car running if hitting the strobes hard with multiple heads, or if only using one or two heads, start up the car every so often to keep the battery from over discharging. You can calculate the drain based on the number of flashes, and get a pretty good idea how often you might need to start the car and let it charge based on the discharge current used.