I worry about this too - I'm in a main office at work and people have always used me as a sounding board. I worry that I have become desensitised to people's problems, but if I have it's only in self-preservation!
don't they have any work to do? YANBU, it's not possible to support that many people, seems rather strange that so many of them are in such a bad way. Choose the one or two who you think really need your generous listening ear and send the rest on their way!