However, I am not sure how much of a real health hazard is presented by occasionally cleaning of your guns, but if it were me I would consider alternatives. At a minimum you should wear gloves to avoid skin contact and use a fan to exhaust the fumes to the outside.
Actually, I remember reading that OSHA recommends only certain types of gloves for trichloroethelene because it can permeate the most common types of safety gloves and then be trapped against your skin by the glove, unable to evaporate. You are supposedly better of using it bare handed because the evaporation is so fast.
I use Eezox for a few older items that I worry about rusting but use it less now that I know what's in it. I don't know that the OP is any danger with his storage situation but I personally wouldn't really want that in my living quarters.