Spend time making yourself comfortable
Sounds simple, but a lot of people treat exercise as a chore, something to be endured. Don't do that! If you don’t want others watching you work out, try to avoid peak hours when there's bound to be a line -- sometimes bordering Disney World status -- for equipment. The least-busy times are during the day and on weekends. Wear something that makes you feel good. It doesn’t need to be top-of-the-line gear, but like in any other situation, the better you feel, the better you’ll perform. Make a playlist that motivates you, or bring a friend to make the time pass easier.
Most importantly, remember that very few people feel 100% confident at the gym. And all those people sweating around you are probably focused on just that. So do the same: put away your cellphone, don't think about work, and take the time for yourself.