Interesting question. I don't think it necessarily means sensitive. I think that women often experience that when they share an experience with another woman, she will be likely to show signs of understanding it by nodding, asking questions, and telling about her own experiences that are related to what the first woman was saying. This way, woman I can react and a kind of mutual understanding comes into place. However, when women tell men things, they often seem at a loss for things to say and end up saying something like "yes, that sucks" or "well, just solve it by doing X", making the woman feel like he doesn't understand or doesn't care. When this happens, men can seem very distant, and it's difficult to establish a connection with them. When this happens it usually makes me lose interest in the guy. I don't bother to try "opening him up" because there are so many nice guys out there you can just talk to. Maybe other women do find it interesting?
Emotionally unavailable means the guy just isn't that interested in a girl.
Emotional available means that a guy/girl is in a time and place in their life where they are able to become invested in a relationship. Examples of people who are not emotionally available are ones who are young and just looking to have fun, recently got out of a relationship with someone they are having a hard time really getting over, or focused on work/school so much that it is their main priority. It also happens when a guy/girl doesn't really like a girl/guy he/she is with, but just with her/him just to be with someone.
Sensitive and emotionally available are totally different things. Because a guy can cry during a romance movie doesn't mean he can be emotionally invested in someone. On the flip side, a guy can be completely emotionally available, and ready to find that special someone, but not care that much if someone's aunt's dog got hit by a car.
It just means that you care about other people. Sometimes, guys are so embroiled with their inner conflicts that they simply have no 'CPU cycles' to spare for other people. And that's just no good is it, as far as a happy relationship is concerned.