There are so many reasons not to charge a car's starter battery in a living space:
battery acid on your clothes when you carry it
hydrogen buildup that could explode
you're going to use an all-metal wrench to remove the battery cables, which makes it easily to accidentally create a short circuit on a device designed to dump a lot of amps quickly
The easiest way to deal with this problem is to jump-start the car and then drive to recharge the battery. You don't have to remove the battery, or risk hydrogen buildup indoors, or get battery acid on your clothes.
Your deeply-drained starter battery is probably sulfated now. It won't be able to hold as much charge, and so won't be able to put out as much cranking current. It's worse when it's cold, so you may find it won't start again the next really cold morning.
Take it to an auto parts place once it's fully charged and they'll test it. The result will be in CCA (cold cranking amps). They can advise you whether you should replace the battery.
There are several different types of chargers out there. Some are pretty dumb, sending a fixed voltage to battery over time. That will do the job of charging up a dead battery, but if you leave it on the charger too long, it can boil the battery fluid. Some even have the ability to supply a lot of amps, so they can be used to jump-start a car if you don't have another car handy.
Smarter chargers have a circuit that moderates the output over time, to bulk-charge the first 80%, and then gently bring it up to full charge, which is something lead-acid batteries like.
Even smarter chargers have a program to de-sulfate a battery, although it takes a long time. Your car would be out of commission in the meantime.
To keep batteries topped off, a smart trickle charger like the Battery Tender is a good choice. I like these for vehicles that side idle a lot of the time.