If I owe my college money can I still apply for fasfa without paying the college back?

I believe you can still apply, no problem. But this problem will come back to haunt you later,
When you're getting ready to get your degree from college, your transcripts will not be released until all debt it paid...

As far as applying for financial aid though, you shouldn't have a problem, good luck!

You can apply all you want to but if you have defaulted on money owed to the government, you may not be eligible for title four aid, which includes some grants, loans, and work study. How do you owe your college money? If you accepted a financial aid refund and then dropped out, you likely owe money to the federal government as well. Please describe the circumstances surrounding your situation and what type, if any financial aid you received through your last school so I can give you the most accurate answer that I can.

In my experience you can apply, certainly, but how/why/what you owe and if you're in default depends on the circumstances. It's a case-by-case decision.