Obviously the blower and controller don’t consume anywhere near the amount of energy that gas delivers, but it consumes more than most people think. The blower on a typical furnace consumes 75-400W per hour of operation, depending on whether or not it’s a single speed or variable speed type.

400 watts per hour isn’t a huge amount of power, a space heater consumes about 1,300 watts per hour, but it’s still a pretty decent chunk of electricity. A deep discharge marine battery (common for cheap backup power) could keep a larger blower running for maybe 3 hours tops, assuming an unreasonable 100% discharge and 100% conversion efficiency. So, not a lot, but still something that you’d need some pretty specialized backup gear to run.

I have an oil furnace, and when I was looking into it it sounded like setting up a battery backup that could allow me to keep the fans running through a power outage like this would probably be more expensive than replacing the oil furnace with an electric one.

If you have to use a battery, then that's probably true. But then again, once you have an electric furnace, the battery requirements are going to go up far more dramatically.

As an anecdote, my furnace blower pulls about 11 amps on 120V AC. So I can run it with a little 2000W inverter generator but won't have much spare capacity for anything else.