Self doubt is healthy. Getting depressed and/or discouraged because of self doubt is not. And your situation is by no means unique. It is VERY hard to catch

Self doubt is healthy. Getting depressed and/or discouraged because of self doubt is not. And your situation is by no means unique. It is VERY hard to catch your first break as a professional programmer. That is just how the industry is, the best way to get around that, is to do what you're doing. You don't HAVE to work for free but if there's no other way it's OK. Work, produce code, make it open source, bring a sample of of your code to your next interview.