I'm in the exact same place right now. I just moved 1600 miles away from home... my family just left this morning and I'm definitely sitting around feeling awfully anxious and miserable right now. But I'm trying to keep in mind how I felt at the interview (amazed, lucky, incredulous to have found a program that was so unbelievably perfect for me), how I felt when I got my acceptance call (over the freakin' moon), how I felt when I came out last month to find an apartment ("hokay mama, leave me here and send my stuff. I don't ever want to leave."), and how I felt when we drove over the mountains and into my new city a few days ago (incredible sense of contentment and feeling that this is exactly where I'm supposed to be).
All of those things are true, so just make sure to keep the confidence and excitement alive in spite of the anxiety. This is a big and scary transition and you're absolutely right, being away from our support systems is way overwhelming. But we always adjust, right? I'm trying to think about all the times when I sat around in an uncomfortable, anxiety-inducing newness and questioned how I'd ever adjust, wondered if it was the right choice, if I could hack it, etc. And now I'm sitting in my new apartment in my new city... so clearly, I surivived and adjusted all those other times, right? This won't be any different. We'll be just fine!
(One other thing that I'm learning... you don't have to feign strength and 100% confidence all the time. If you feel crappy, then you feel crappy. Acknowledge it, experience it, and then move on. The worst thing to do, I've learned, is to sit around tightly holding in your anxiety -- if you need a good cry, have a good cry. Then call someone you love or go out and do something fun! Give yourself permission to feel badly for awhile if you need to -- pretending that you're totally fine and/or feeling like you shouldn't feel upset or anxious will, in my experience, just make you feel worse. Again, let it be what it is and then take a deep breath and move on.)
If you need to commiserate, I'm always good for some PM therapy. 😉