Congratulations! You are an insightful person, when you want to be. That means there is real hope for you.

People aren't going to like you when you don't like people. You have judged everyone around you as being too superficial to see past externalities. Why should they want to know you? You think so poorly of them.

There are fat people who do have friends. And everyone in the world is not totally superficial. You don't know what people are like because you haven't gotten to know hardly any people.

So make the most of this new connection you've established. Then start telling yourself that you can have a circle of friends, if - and only if - you are willing to invest a heck of a lot more effort than you've been putting out. And, yes, that's hard to do when you are depressed - very hard. I totally get that. Plus, in the beginning, you won't be too good at it. You're way behind on the learning curve. And you'll get some rejections. So you move on and try other people.

It took some talent to even make it in to the graduate program you are in. If your instructors are losing faith in you, it's because they don't see you putting forth the requisite effort. Graduate work is very hard. (I know because I tried it for a year and packed it in.) But ask yourself, "What are you going to do instead?"

So decide that life is all about working for what you want and trying and trying and more trying. Be glad you're not on an assembly line somewhere doing some boring-@$$ job with no hope of your opportunities ever changing.

Another thing: It is more and more true today that, in order to get anywhere, you need people to vouch for you. So start doing whatever you can to impress those professors and instructors that you at least have some interest and enthusiasm for the research they are supervising you in. Right now, you sound like you don't have much interest in anything. No one wants to hang around with a person like that, no matter how thin you might be. In order to be interesting, you have be interested. That gets built on a lot of tedious work.

See if you can be useful doing some tutoring of struggling undergraduates. It can be confidence building to help someone less able than yourself.