Will having debt disqualify me for the Navy?

It's possible. Debt can deny you a security clearance which will dq you from alot of jobs, this will not be to the best interest of the navy and they can dq you because of that. Debt denies you from a clearance because you hypothetically will be tempted to sell information to pay off your debt. I don't think you'll have to worry about that though, it's only for people with massive amount of debt and horrible problems like you mentioned. Ask your recruiter to be sure.