There's been a lot of talk about "Stay" on this forum before, and it seems like a lot of these ideas are similar to those. I think there's an overall meaning that many fans have reached at least a general consensus on.
Clearly the character in the song has suffered at the hands of some sort of abuse; many assume it's a woman who's abused by her lover, though it could just as easily be an abused man. We get the impression this person (we'll just say 'she') is a human wreck wandering the city.
It suggests the whole psychology and emotional cycle behind domestic violence. It's very common for women to stay with an abusive partner--it's practically a stereotype. A woman who suffers this kind of abuse believes that the violence is her fault and actually has an inability to place the blame elsewhere, mainly on the husband. There are even some experts who believe that women with a certain dysfunction go so far as to assign meaning to the abuse (He hits me because he loves me... When he hurts me, I feel alive.) Clearly we can sympathize
with a person in this situation.
One key line--"a vampire or a victim"--is what makes the song appropriately complex. Many of you mentioned you know people like the one in the song. We all do. And we know how exasperating it is to extend our sympathy and support to those people and have them turn away and again do the wrong thing. They drain the love from deserving people, and give it to the undeserving. It depends on who's around.
The songs takes a turn at this point--there's another character, 'Me.' You look through me, talk at me and when I touch you you don't feel a thing. Here's where Wim's movie (and his other one, Wings of Desire, for that matter) comes in. In them, angels zip around the world (up with the static) watching episodes of human happiness and sadness like television channels. But like the TV viewer, they are unable to intervene in the lives of humans, whom they love. They're unseen, unheard and unfelt. This is a frustration many real people, including Bono obviously, can relate too.
Without giving away the movies (not that this would) the situation changes for an angel, as hinted at the end of the song. I have an entire other theory on the personal significance of that and how it relates to Zoo TV and Pop, but I'll hush up unless anyone wants to hear it.