So I can think of three ways around the whole, 'how do I get experience, when everyone wants some first' paradox. There are probably others.
1) Know someone already working there who can help get your portfolio past the gatekeepers.
2) Apply anyway, with a kickass portfolio and hope someone notices.
3) Do some freelance work on smaller projects, these usually avoid the more checkbox oriented gatekeepers, and they totally count as viable experience. This doesn't even need to be paid work, but having that experience of working on and with a team can be huge. Also this helps lead back to #1, as folks you work with might be able to help get you in contact with others who are looking.