In the US, to be a lawyer you have to go to a 4 year college and then obtain a law degree (usually 3 years). After that, you have to pass a state bar exam. There are a handful of states that allow you to skip law school, apprentice in a law office, and then take the bar exam to become a lawyer; I think California may be one.
One clarification to your question. When people refer to "contract lawyers" they usually mean lawyers that work under temporary contracts on a project-by-project basis. They differ from the usual associate and partner lawyers who work in a law firm. I think what you meant to say is that you want to be a lawyer that works on contracts; this is sometimes referred to as a corporate lawyer or transactional lawyer. These lawyers negotiate contracts, draft contracts, and deal with disputes that come out of contracts. In law school, every student takes a contract class in their first year to learn about how contracts work and how to interpret them.
My firm has represented clients all over the country for 17 years and you will be represented by a lawyer...unlike other firms based in New York who hire non-attorney reps.
Personally at the firm I have probably won over 50 cases in the last year alone. Each case is different however. Feel free to call my office for a consultation during normal central time business hours. Give your information to one of my intake ladies or gentlemen and an attorney will review it for you. We won't take your case unless we think there is a chance you can win.
You don't have to prove you can't do anything, only that you can't sustain a forty hour work week at certain levels of exertion in connection with your past relevant work, your age, and your education. Depending on your age this might be easier.
Feel free to peruse my other SS answers on this site.