Strength Training w/o a gym. Is it even possible?

I do strength training from home. You can use free weights and a bench if you can find one at a garage sale or on craigslist. I found one at a garage sale for $20. Or you can do a lot of strength training using only your own body weight. There are lots of things on the internet for free that you can try.

You can start with body weight exercises too and look for dumbbells or barbells at yard sales if money is tight. Just google Beginner body weight exercises and there are several sites that can help you get started. Good Luck on your journey!
Edited to add: I like Jimmmer's answer better!

There are a ton of body weight exercises you can do to start. Eventually you'll have to change the leverage of how they're done because as you lose weight, then the resistance from say a regular push up will decrease.
One of the better pieces of equipment I tell people to get who don't have access to a gym, travel a lot, or just like working out at home is a TRX SUSPENSION system. Well worth the money spent based on how many exercises you can perform on it, how much room it takes up, what it weighs and how you can keep increasing intensity. And it's less than $200 (I've seen it for under $150 and on Craigslist for even less)

A.C.E. Certified Personal/Group FitnessTrainer
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Been in fitness for 30 years and have studied kinesiology and nutrition

Body weight exercises are great such as push-ups, pull-ups and dips.

You can even fill up 1 gallon milk jugs with sand (or something denser for more weight). Then maybe get some 5 gallon buckets filled with sand and pad up the handles. Get innovative. Who says you need expensive weights?

If you can spare a little cash you can get the Bowflex Selecttech interchangable dumbell set. Its basically like 20 dumbells in one where you can select the weight you want with a dial. There are other companies that make similar porducts too. Google it.

Bodyweight exercises. You are your Own Gym is about $10 on Amazon. Later you can add a door way pull up bar (Iron Gym) for less than $30.

youtube.com has many workout videos for free, weights on craiglist

Everyone already beat me to the punch about the body weight fitness. I think for beginners who aren't packing a lot of muscle to begin with, body weight fitness is the easiest way to go. Just push yourself as much as you can, within safe limits, every other day and your muscles should grow just as they would at a commercial gym. Your body doesn't know the difference between pushing 100 lbs of iron and pushing 100 lbs of flesh.

Another benefit is that you can let yourself go when you work out in privacy. You might not want to go to failure in front of others at a gym, but it wouldn't really be a problem if you were alone. Of course, only you would be able to keep yourself accountable as well.

You can also get a decent workout with resistance bands.

ashotofadrenaline.net

This one hasn't been mentioned yet. Free eBook, tons of workouts. The options are endless with working out at home or with no weights. I'm taking a break from heavy lifting and doing bodyweight only. It's serious business!

You can find a pull up bar lots of places, other than Wal-Mart. I found one at Aldi (grocery store) for $12 not long ago. It will be a good investment to have in your home during rainy or cold seasons when you can't get outside to be creative with a pull up bar.

I'm doing the You Are Your Own Gym program. I haven't read the book but the smart phone app is $3 and so easy to use.