It doesn't matter, as they cannot ask the nature of your disability. I've rented apartments before and I tell them I'm disabled and on social security. I just say "I have a medical condition that prevents me from working". And for the apartments I've rented, they didn't ask any further questions. Their main concern is making sure you can afford the apartment rent. My income wasn't high enough alone to rent (it had to be 3.5 times the rent), so I had a family member consign for me. Just make sure you look clean-cut, wear a nice outfit and don't look scruffy.

Renting a room is another story, people are generally more concerned with details about things because often times you're living in close proximity to people in someone's home and you're sharing a lot of the same space...and they want to make sure everything will go okay, that they don't get someone who wouldn't fit in with their lifestyle, or someone who will trash the house--throw loud parties, does illegal drugs, etc.

As for your neighbors...you don't have to tell them anything. I hardly ever saw my neighbors. It's none of their business. And if they're nosey, make something up...they're not going to find out it's not the truth. And you can always say you're in school (I sometimes say I've been working and now I'm going back to school to finish my degree---and I'm not a traditional student age either).