I know just how hard it is to reach out. I only have 2 really close friends and I still don't even feel comfortable bringing my issues to them. It sucks, but all we can do is power through to prove the doubters wrong!

You are in your 20s, if I recall. You ARE compatable to other humans But, it sounds like your expectations are high and there might be a little (more than a little?) lack of confidence. I had a rough go in my 20s and back then (30 years!) there was no internet to turn to. I hope you'll use this community to find the support you need to get through a busy graduate program. Let the graduate program be what and how it is, a means to an end in which you are laying the foundations for a great future. Don't try to get milk from a stone there. If it's not there, it's not there. Connectivity will come in life once you start letting your barriers and expectations down. That's not work you need to do now when you're already carrying a big load. For now, just be who you are and succeed at your studies and breathe.

I wish I could advise something, but I am in a similar situation. I like my chaos on the inside. I do salsa classes and socials if I want people but don't want to talk. Seems to take care of things for me. I am also big on being a loner

It's very relatable. when I was young I always tried to fit in.

I remember one time on school camp where my group of 15 plus 3 teachers rowed out towards an island for our "ship wreck scenario" activity. On the days leading up to it I had not gotten on well with the group, only 2 of them realized that I was an experienced camper and navigator and the rest didn't care. When I offered to navigate across the 22km of hills and valleys they told me to get lost and they followed goat tracks instead. When I tried to tell them we were going the wrong way they told me I was wrong even though the teachers had stopped walking 50m bahind us. When I tried to help one of the weaker students trailing at the back they told me to hurry up. So at that point I was pretty tired of it. I thought "I'd be better off on my own" but then I remembered how I had accidentally spilt half my water on the first day and the weaker guy had given me some of his water even though he didn't have to.

It made me happy and I became pretty good friends with him.

The point is, even if you feel like you're alone and things get tough from time to time. Being kind to a stranger for no reason can end up making you a good friend.

Evidently we don't really talk anymore but having a friend for those extra 9 days made the camp a lot more fun, even though I injured my leg and it became septic with infaction... that wasn't fun especially when we were swimming with the sharks.

I wanted friends for about a minute, when I was very young.

I learned quickly how people are, and realized I didn't want to be friends with 90% of them after all. I just wanted to be overlooked or ignored, so they'd stop picking on me.

Then I grew up, left school, found my "sea legs" as an adult--and it got even worse!

People are terrible. You're not missing much.

Ah hell... I'm always looking to make new friends. Y'all come hang out....

Feeling inadequate is commonplace in my life. It feels like I'm doing all the rights things but still nothing seems to happen.

Probably.
Not I, though, as I can take people or leave them. I enjoy them when they're around but afterward I need to recharge by going outside and being one with nature.
Usually it's my friends trying to persuade me to do something with them, and I'm arguing because I just want to be off somewhere reading a book.

I've been there. I tried dating locally`but I can't lower my standards to date a Trump Supporter, anti-immigration person or tobacco smoker. Its a connundrum.