This is the $64,000 question, isn't it! Speaking from my own experience, I've found that post houses will grant you an internship if you have the drive to stick it out. Many people desire to be there and say they'll do whatever it takes, but when it comes right down to it only a few will see the value and persevere.
Internships are not neat and tidy. Often times they are unpaid. Most of the time interns are making coffee, cleaning the place, answering the phones and running lunches. And if they do those things very well, then they get the privilege of sitting in on sessions and learning the trade. Personally, I think this is an excellent method; not only do you weed out the folks who don't really want it, but you also create a character-building experience for the people who do.
Apprenticeships at larger facilities (like movie studios) are a different subject, since you're involving parties like unions and guilds, but the payoff is potentially huge. Landing a seat next to an A-list feature film sound designer - priceless!
How do you secure these positions? Ask the folks around you for introductions. Draft a resumé and personally deliver it to all the facilities you are interested in. Buy lunches for mangers and studio owners. Buy lunches for the creative types that work there. Bring some homemade cookies - whatever gets the point across that you really want this.
Another thing to note is that it's not all about the tech savvy skills; personality is king. If you can't get along with people or have an ego, then you may as well throw in the towel right now because no one wants to be around somebody like that, especially when they're first getting started in the biz. (If I had a nickel for every 4-year grad with a degree in music tech who moaned about making coffee…)
The most important thing to remember is this: Be realistic. Success will NOT happen overnight. Find the most creative people at a facility and court them. Befriend them. Pursue your passion with after-hours and weekend projects, but earn your paycheck by answering phones. Be committed - and don't give up!