I wore my binder for a few hours too long the other day and for about two days since I've had a bad ache in the left side of my chest. I'm worried I managed to damage something, but I can't afford to see a doctor. How can I help the pain/ make sure it's nothing serious?

Look up free or low cost health resources in your area. You do need to see a doctor. Best case, it will go away on it's own with time. Worst case, you've cracked or bruised something internal. The only sure way to know is to have a doctor examine you.

In the mean time, you should not be binding. General painkillers will probably take the edge off. But you're just treating symptoms and not the root cause until you get checked out.