I understand anxiety, I am out of a job right now because my anxiety is so crippling and I'm 22. Even going to college makes me incredibly nervous. I know it's hard to not care, but that's literally what everyone else in that environment is doing actively. I don't really know any secret tricks to stop caring because I would have used them myself. Perhaps medication is the answer. It's something I'm going to look into. But also, you could look for jobs that are "behind the scenes" that do not involve working with people, or "customer service".
There are places like mail rooms, libraries, kitchens (dishwasher or prep cook or line cook or baker - bakers actually work night shifts a lot of the time and they are often working alone). There are telephone jobs that only require you to talk on the phone and never be seen, such as 911 operators or sales reps. You could work in a hospital in jobs that are in the labs, or environmental services, or technicians that only work in one particular part of the hospital and never meet patients.
Just look around your town for all the places you would want to work at. Then inquire about the jobs that are more "behind the scenes" type jobs and not customer service jobs.
Unfortunately being young, people just assume the only jobs out there are customer service related jobs, but that's not true. There are lots of jobs out there that don't require public interaction but they are often harder to find simply because they are less known.
You could also go to a technical school and get a certificate in a hospital job.
You could volunteer at the hospital first to get your foot in the door. They have departments like medical billing, patient transport, labs, cleaning, cafeteria, laundry, etc.