If my gym is closing down due to bankruptcy, can/how do I get my money back?

Effectively you do not have any options is the gym is actually filing bankruptcy: not only does bankruptcy temporarily (for months, or even years) "stay," or stop, any collections effects, but typically, creditors only get dimes or pennies on the dollar from the bankruptcy process (and sometimes nothing at all). For $300, it is doubtfuly that it is worthwhile to try to put in a claim to the bankruptcy court.

If the gym is not actually filing bankruptcy but is simply going out of business due to insolvency, you could try suing to recover your money. If the gym is an LLC or corporation, again, though, it's probably not worthwhile--if it is a LLC or coporation, you could only sue the business itself, so if the business has no more assets or income, there's no way to recover money; the owner can safely walk away from a debt like this, because he is not personally liable.

If the gym is a sole proprietorship or partnership (not an LLC or corporation), you may be able to sue the owner(s) personally, since there  is no separate legal entity--the owner(s) are the business. In that case, it's probably worthwhile to represent yourself in filing a small claims case, since the fact that you could go after the owner(s) personal assets may give  you enough leverage that you can collect something.