Rachel,
While it is true that many entry level positions are disappearing in favor of internships I wouldn't discount the value and work experience you get from those internships. My first position was part of a work study program where I worked underneath a curator. I did condition reports, handled artifacts, entered data into the museum database and assisted researchers - all of which were very valuable skills. My second position was as an archivist and it was a fellowship as part of my graduate program.
Often, if you are paid by an organization you can negotiate a title that better reflects the work that you perform. I was able to call myself an "Assistant Archivist" instead of a "Student Intern" as I was paid for the hours that I worked (20 hours part time) and the duties I performed were almost exactly the same as other senior archivists. I processed collections, created finding aids, scanned photographs and curated small exhibits.
When I was hired as a full time Registrar it was because I had two years experience working under a curator performing registration tasks and two years working as an archivist where I learned valuable skills working with manuscripts and photographs. Most senior members in the field understand the path that emerging professionals take and look at what you've accomplished. Of course, networking is a big part of it too and volunteering and interning help build a network of professionals that can assist you in the future.
Its a tough field to break into but with some hard work its not impossible. Good luck!