Catch 22

As I was searching through countless job postings I noticed that recently many of the jobs I was looking at that I would be really interested in required 5-7 years of experience.  It seems like a classic Catch 22 situation to me.  I didn't have 5-7 years experience because I couldn't find a job.  But how was I supposed to get the experience when I couldn't find a job because no one would hire me because I didn't have enough experience.  It just sort of goes round and round.  I was wondering if recent graduates or people thinking about changing careers are coming across the same roadblock?  Is this happening to you too?  How do you overcome the lack of experience problem?