Can you become a chef without cooking school?
Yes, you can be a chef without going to cooking school, there are plenty of great chefs who never had any formal training. Paul Prudhomme, Mario Batalli, Tom Coliccio, and Charlie Trotter are some famous chefs who never darkened the door of a cooking school and those are just the ones I can think of. Some top chefs have been known to say they think its a waste of time, I'm not sure I agree but you get the idea. One thing that is pretty much agreed is that the real training starts from when you get your first restaurant job. One story I heard from a top chef (I can't remember which now) was that he told the chef in the restaurant where he worked that he was thinking about going to cooking school and the guy just looked at him and then handed him a bunch of onions and a knife - he never did go to cooking school. 

All you need to become a chef is the willingness to work very hard and long hours, and some skills. And restaurants to work in, of course. The skills you can learn at home, and much of that would be food preparation as that's where many chefs get their start. Learn to prepare meat and vegetables in chef-y ways, and learn to be consistent. Practice lots of knife skills - that alone can get you the first job. There are many books out there, and lots of tutorials on the internet for that.