Hi, all I can say is yep I know the abyss, and sympathies. I've lived on basically fast food for days at a time. Tip: freeze. If you do have a good day and cook, make twice as much as you need and freeze the extra for another day.
Try to wash and get dressed, even in your comfiest tracksuit or whatever...I find it does help me feel better to be (relatively) clean...slobbing around makes me feel even more crap about myself. I know it's hard.
Excercise definitely helps me. I used to attempt to go to the gym but yeah, if you're depressed, seing all the fit toned people is just more depressing...and working the machines (or failing to), ugh. I now run. If I can't motivate myself to go for a run, I just go for a walk, even if it's only 5 mins to the shops and back...it is still exercise, and clears my head a bit. If I can't even be bothered to do that, I put on cheesy music and dance around like a freak or do housework to it.
I know, it's not that easy. I'm not trying to sound like a ray of sunshine, cos I have been where you are.