Following on from the discussion with WBSH et al, I will add some of my own thoughts in here, especially since there have recently been positive developments in this area.
In my experience, one of the main issues with these relationships are knowing where the line stands between platonic and sexual love (in theory it is easy; a platonic relationship has no sexual involvement, a sexual relationship has at least some, but in practice it is complicated). With regards sex and love being different things, I think it would help people understand if more analogies were made with "family" relationships, e.g. someone might love a platonic friend as they would love their mother, or a close sibling, neither of which are usually associated with sexual involvement.
The bigger issue in my experience has been third parties spreading nasty rumours about the male constituent of the friendship and undermining the friendship. This kind of thing is hard to deal with due to the social norm that whenever a third party interferes with a friendship, you assume the friend 100% guilty, and the third party 100% innocent. Thus, if either of the friends go to anyone for advice on how to deal with this third party, they will be told that they are being shat on by each other. There's also the social norm that if a man is accused of abusing a woman, you believe the accusations, thus people tend to believe whatever rumours are spread.
The issue of a "true" friend is double-edged here, as when the friendship is broken up, the male constituent usually ends up in trouble for what he is rumoured to have done. The truer the friend, the more trouble will be required to break up the friendship and thus the more likely it is to survive, but also, the more trouble the male constituent will end up in if the friendship falls apart. The concept of a friend who would stand by someone "no matter what" is unrealistic, e.g. if someone is made to choose between one person or another (which can happen in these situations) then he/she can't be a "true" friend to both of them. Again, an analogy with "family" makes this more clear; this stipulation of "stand by you no matter what" is rarely if ever applied to "family".
I think the social segragation of the sexes also makes platonic relationships difficult, particularly for men, as male shows of care and affection are often misinterpreted as being sexual. Like-minded constituents of a platonic relationship may violate gender norms and be put under pressure to conform, which causes them to be pulled apart as they lose the things they have in common. In general platonic relationships seem to work best if they occur within a "family", mainly because society considers "family" to be trustworthy and "friends" to be unimportant, so "family" are given a lot more leeway (plus, "family" relationships are normally assumed asexual by definition) so third parties are less likely to take offensive action (and less likely to get away with it if they do).
In the past couple of years, however, I have been able to form some platonic relationships with women without encountering the above issues, which has helped my confidence in them and shown me that it is possible to make them work. They are becoming more acceptable in society than they used to be, but as long as gender and "family vs friends" stereotypes continue to be accepted without question and enforced by society, there will always be barriers to them being accepted as they are.