Apple would have to borrow heavily to buy itself back, just like every other company that buys itself back. But if any company could go borrow $400bn in cash, it's Apple.
I wasn't considering the extreme pedantry that I'm apparently receiving when I made the post. I was using the phrase "buy itself back" as going private. The ownership structure after which point that it is no longer a public company was irrelevant to me at the time I made that comment.
The company can't "buy itself" into a private company either. It doesn't work for a whole host of reasons, one of which being that they aren't sitting on the ~600 billion in cash that would require, another being that buying a bunch of stock back is fundamentally not the same thing as being a private company. But also, if you buy all the stock back, it means by definition that you have liquidated all value in the company (or else we're assuming that the market undervalued the company so much that it thought nothing but the cash on hand was valuable).
If you assume that Apple is going to borrow the 600 billion to buy all its stock, then you are effectively saying that whoever's left holding the "private" company is holding a 600 billion dollar loan. I do not share your optimism that anyone can get a loan large enough to buy Apple.
Apple can't take itself private unto itself, there must be a human owner or owners that own all of Apple stock as a private corporation.
Apple could team up with JP Morgan, Silver Lake Partners, and a dozen other firms to take itself private however, and they along with a likely small class of shareholders would own the private corporation (eg perhaps Tim Cook and Laurene Jobs would would choose to retain their large positions).
There is no possible scenario under which a corporation can exist and continue to exist indefinitely in any of the 50 US states without a human owner somewhere in the chain, even if said owner is buried under a dozen shell companies.