I dated a guy just like this! He went and entire summer without washing his clothes, and to make it worse, because of his poor self image, he would wear flannel shirts over his T-shirts - IN THE SUMMER! Towards the end of our 5 year relationship, sex was minimal. During the last 10 months of the relationship, we only had sex once, and that was because I was moving out of state to go to grad school.
He was depressed and he knew it, but didn't want to do anything about it. He didn't like the idea of meds or therapy so when I moved out of state, I ended the relationship a month later. He did eventually get over the depression, but it lasted at least 5-8 years.
Personally, I would opening confront him on it. I wouldn't doubt if he knows he is depressed, but doesn't know what to do about it. I'm seriously surprised that no one at work has called him out on it. Maybe someone has, but he's too embarrassed to talk about it.