a 2000 watt generator will run up to about 16 amps (2000 watts divided by 120 volts) which is actually pretty sizable. That might work for you but you would be pushing it.
I just purchased a 3200 watt (continuous), 4000 watt peak generator from Harbor Freight for the home as it was on-sale for $339. It actually is a nice generator and not too terribly loud. I plan on using it for power outages to run two refrigerators, my furnace blower motor, a sump pump, lights and a TV.
Here's a link:
Harbor Freight 4000 Watt Generator