From what I have been told by at least two people either currently in service or recently out of service (both, oddly, in Army Intelligence) quite possibly the best way to stay out or get out if you're already in is to say that you're homosexual. While I'm not totally certain of how likely this is to actually work it's an interesting concept nonetheless. Basically you have a condition that cannot be tested for or otherwise proven effectively and objectively. You could ask friends and relatives, but they're just as likely to lie on behalf of the person. You can't very well claim that they need to have been in a homosexual relationship as they could very well never have found the right person for them.
Among the benefits of this is that it's legal. You're just getting the government to say that they don't want you whether you're otherwise fit to serve or not. A discharge on these grounds is apparently a general discharge which reverts to an honorable discharge following six months without incident. Apparently being classed as a general discharge it also won't have any effect on any sort of permanent record.
I have to admit I'm not totally certain if it would or wouldn't work. I've spoken to people who have served and they all seem to agree that it would, but that's not exactly official. Still, when three or four Arab language specialists were suddenly discharged under these guidelines in the winter of 2002 (or so I was told) it seems like at least a few people are using it as a way out.
Update: I have since been told that this does not work in times of war. Try it during peacetime though to get out of the army. But for avoiding a draft? Not bloody likely.
Second update: with the repeal of "Don't Ask, Don't Tell" and the significant victory of being able to finally serve openly this is no longer any sort of option. It's being retained here for historic reasons since for quite a while this was a valid method that some people employed.