Start off by walking, a lot. Get yourself to a point where you can sustain physical activity for awhile so at least you won't be wheezing and feel like shit while there.

Then, pick a few exercises you are confident you can do competently and just focus on those for a few weeks in the gym. Slowly add in more things and update your regimen over time.

If you just go in and focus on your stuff you will probably hardly notice the other people.

Also, different times of the day have different people. If it's not the summer break the daytime has no teenagers, and the people there will be the most dedicated and serious. They won't care at all about you.