From an OHS point of view... the gym does have a duty of care to ensure that people entering and using their facility are not exposed to undue risk as a result of the facility's equipment, processes, etc.
This duty of care though is limited to that which is "reasonable" for them to (A) exert some control over; and, (B) have reasonable knowledge of. ( This doesnt mean that "ignorance is bliss" - quite the contrary, the test is not "what you know" but "what you SHOULD know" )
In this case, it is not reasonable to expect the staff or owners of a gym to adequately diagnose and manage health conditions such as anorexia. They are not health care professionals. If they WERE to intervene based on their own, unqualified, assessment then it is likely they would open up a far greater can of worms than if they just left it alone.