A facility with exercise equipment and sometimes instructors, open to the public or those on some membership list
{i} establishment providing means to improve fitness and health (exercise equipment, pool, sauna, massage, etc.)
a place of business with equipment and facilities for exercising and improving physical fitness
A health club is a private club that people go to in order to do exercise and have beauty treatments. a place where people who have paid to become members can go to do physical exercise