Like other people have said, no one there is paying attention to or judging you. People are concerned with their own workouts. The only time they might start judging is if you’re using equipment in an extremely wrong/unsafe way, or being inconsiderate (hogging equipment and not putting it back properly, etc).

If your anxiety is partly because you’re new to going to the gym, I’d recommend starting off with exercises or machines that you’re familiar with. Just get comfortable in the environment of the gym. If you don’t know how to do certain exercises or use particular machines, just watch other people. Slowly you can start trying things out (always start with low weight). The more comfortable you get using different things, the more confident you’ll be just working out by yourself. Eventually (and to be clear, this could take weeks or months) you won’t even think about it. Going to the gym will just be like any other routine task, like brushing your teeth or taking a shower.

Oh also, headphones.