Every damn time this topic comes up, all I can think of are really paranoid conclusions given my current situation.
Important Note: I am not, nor have I ever been, some variety of sex symbol and doubt I have the potential to become one. If I did have such potential, I would be using it to become rich, famous, and have various lesser humans doing my every bidding while I do something horribly embarrassing for all the world to see (but I wouldn't care, because drugs).
I am female. I have, for most of my life, had mostly male friends. This is because I have a lot of interest in things like video games, collectible card games (mostly Magic: the Gathering), fantasy movies/books/TV shows, science fiction, roleplaying, anime, etc. As it was, and sort of still is, these activities were/are mostly enjoyed by males. And also me. So, in order to talk about 95% of things I like without weirding someone out, I had to be friends with guys. This only got worse when I went to college and picked a major with a very tiny number of females in it, leaving me with almost no female friends (and those I am friends with share some of those interests).
So, given all of that, I see a small number of options (aforementioned paranoid conclusions) given the statement "it is not possible for men to be 'just' friends with women":
1) I am sexually desired by every man I am friends with, to the point that our friendship will end because of it (see: I do not think every man is reduced to a slavering sex beast when slightly attracted to any one given female, and I see no need to end any friendship because a dude had a vague crush on me at one point). I am personally not so narcissistic to think this is true, because that is certifiably insane. This is not to say I am friends with a ton of people, but I'm pretty sure that, unless I'm a sex god, I can't be that desirable to every random man I befriend. That is statistically impossible given my current level of attractiveness and the fact that you can't actually photoshop reality (yet. Lookin' at you, Google glasses).
2) I have (virtually) no friends. Every man who claims friendship with me secretly dislikes me to the point of not being attracted to me. Perhaps they regard me as some form of subhuman, and being attracted to me would be like wanting sex with a dog.
3) Along the lines of #2, but: I am not as close to my friends as I think I am. Maybe they invite me places to feel better about themselves ("look me, I friended a girl! Aren't I awesome and totally progressive?") or hang out with me until they can hang out with my husband (who works weird hours sometimes). Perhaps I am just someone they put up with while their real friends are busy. If this is true, I'd imagine them eventually winding up at #2, since I like to talk to my friends a lot.
If men CAN be friends with women? I'd imagine that all of my friends are actually interested in hanging out with me and not secretly sick of me, improbably wildly in love with me, or bored with my existence. Then again, I don't think small amounts of sexual attraction are friendship-ruiners (I am not the thought police, and if I was, a lot of people would probably be arrested for much worse). It's only when we hit unrequited love that this becomes an issue, and only if you can't get over the person you're in love with (yes, it is possible to be "friend-zoned" and eventually become ok with that and move on with your life).