I live in a cold climate - tonight's forecast is -37C (-35F). Your main problem is that you don't have a car. Most people have absolutely no conception of what an impediment this is in a cold climate:
Could you maybe join a gym with a friend and have him/her give you a ride? A scheduled class would be your best bet - I do Cage Fitness and Karate. If you could get out to a class a few times you might even make a new friend who can give you a ride.I can be outside in a light jacket, shirt and vest in 15° if I am skiing.
You're going want to buy a doorway pull-up bar and some free weights for working out in the house. If I was going to exercise exclusively on my own I would probably also look at some workout tapes, otherwise it gets too easy to cut yourself slack.