Hey Megan,
I think the largest problem here is that you're in Los Angeles. You can't go an hour or two without someone looking for an intern for their documentary on any of the east coast (or, hell, San Francisco) craigslist tv/flim/video posts. Los Angeles... well, not so much. So, unlike show business, you don't really break into documentary. You can find a company that produces documentary programming and try to work with them for free and try to get a sense of what they're doing. Or more probably, you can hone the various skills that make up a good documentarian - storytelling, attention to detail, fundraising, stupid tenacity in the face of overwhelming odds, good editorial skills, and how to use a camera under less than ideal circumstances - all of which are skills which can be picked up either (as John notes) by grabbing a camera and exploring your own ideas... or, quite frankly, in the service of any aspect of production in whatever your area might give as an opportunity (and LA has its fair share, to be sure). And if you happen to spy someone posting on craigslist or mandy.com looking for an intern for their documentary, you'll be able to apply to that with some amount of skill that you've been working on developing. It's not a great answer, but perseverance is probably the biggest chunk of the recipe. But that's kind of true for anything worth doing...
