a dilemma or difficulty from which there is no escape because of mutually conflicting or dependent conditions
The classic statement of this situation is in Joseph Heller's novel Catch-22 (1961) from which the expression is taken : Orr would be crazy to fly more missions and sane if he didn't. But if he was sane he had to fly them. If he flew them he was crazy and didn't have to but if he didn't want to, he was sane and had to.
1997 - New Scientist - It's a catch-22 situation…you cannot get the job without having the relevant experience and you cannot get the experience without having first done the job.