My grandmother told me when I was a kid that men and women can't just be friends. Like so many, I suffered from the idealism of youth and didn't want to believe her. Now, I think that it's pretty much true, unless the friends are so hideous, they couldn't imagine sleeping together.
But let's look at the General Social Survey to see how often we see cross-gender best friends. The top graph shows what male respondents had to say about who their best friend is--other than their current partner. I've also broken it down in terms of marital status. I've done the same thing for women in the lower graph.
For men, having a girl as a best friend is quite rare for married men and never married men (8.5% and 10.4%, respectively) but it's close to 20% for separated, divorced, and widowed guys. Younger guys have more (male) friends than older guys, which might explain the low number of female best friends among the never-married. Older men who have been through a breakup or the death of a spouse probably don't have a group of pals anymore and so are more likely to make a close friendship with a women.
Still, I'm a bit skeptical of at least some of these friendships--I imagine one or both friends have thoughts of something more, but don't report it because it hasn't developed that far yet. And of course, cross-sex friendships are rare among married guys. What wife with any sense would let her husband get socially intimate with another woman?
It's the same for married women: only 3.1% have a man as a best friend. Hmm, isn't that estimate also about the same as the number of gay men in the population? It's doubtful that a husband would allow any other type of close male friend. It's also rare for widowed women: perhaps that is due to the fact that men are comparatively scarce among older people. It's a little more common among separated and divorced women. Fourteen percent of never-married girls have a male best-friend. Once again, I wonder if some of these are possible future relationships or one-sided romantic/sexual ones.
Basically, I think Grandma was right.