I think it would be grammatically correct to say that the street is "getting crossed" when someone is currently crossing it, although it would be better to say that it is "being crossed".
A Street Being Crossed vs A Person Getting Dressed
I think the difference, when compared to getting dressed, is that being 'dressed' is a persistent state. Once I am dressed, I stay dressed until I undress. How does a street become 'uncrossed'? Instead, the street is either being crossed right now, or it isn't. It won't stay 'crossed' after someone crosses it. I mean, technically it is a street that has been crossed, and therefore could be describe as 'crossed' forevermore, but that's not a useful concept.
Once again, compare this to getting dressed. You could say that one is "being dressed", but that means that clothes are currently being placed upon them, not that they have had clothes on for some time and still do. For that persistent state, or condition, we say that they "are dressed". In the process of "being dressed", they "got dressed", and now "are dressed".
Crossing a Person
To cross a person means to upset them. If someone stole my red stapler, you could say that they crossed me. One would not however say "I am crossed". One would say that "I have been crossed" and that "I am cross". This is another situation, rather than a condition, so we don't usually use 'get' here either. In other words, one would not normally say "when I get crossed", but instead "when I am crossed".
Other Examples of "get [verb]-ed"
In all of these, "get" means "become".