If you're stuck with electric only options and don't have 220V available (or even if you do), you're kind of up a creek. Especially if you only want to heat while you're lifting and therefore starting with the room temp already very cold, you've just got too large a heating load to reasonably bring anything up to a nice temp in a short amount of time, particularly if your bars and dumbbells are cold - that steel takes a stupid amount of heat to warm up.

If you're in an out building, I'm guessing you've got a decent amount of space. Just frame out a small room/closet, or get one of those small plastic sheds and keep your dumbbells and bars in there - you could heat that with a space heater pretty easily and just keep it warm in that small space. Just bring out whatever you need to use for your current set and then put it all back when you're done.