I own and operate an internet only talk radio station that doesn't use any copyrighted music, only royalty free "podcast safe" music. When you purchase the music, it licenses you to use in a podcast, on-air or streaming over the web.
I make a small profit, not quitting the day job anytime soon, but it takes a combination of things to do this. Website advertising, selling audio commercials to customers who want to reach a national audience and running PPI (Pay Per Inquiry) advertising for national companies (Dish, Lumber Liquidators, etc.). If a listener calls using your toll free number, you get paid. If nobody calls, you get nothing and it cost the advertiser nothing.
Sooo....you can make a profit, but I'm certain that if I were paying music licensing fees as your typical music station, the odds of making a profit in that scenario would be slim and none.