There has been a selective service agency since 1917 which controls entry to military service in a codified way. There is a classification system in the US CFR-32 which lays out the different categories a person may fall into.

The category of "4F" is reserved for those physically unfit to serve. The criteria for being unfit are chronic conditions or physically disabilities which can not be fixed through military training. Things such as epilepsy, blindness, missing of a big toe, and so on. Merely being unhealthy in terms of being overweight is not a disqualifier for the draft , since that is something which can be fixed through training.

The men considered 4F are rejected at Military Entry and Processing Selection (MEPS), their status is recorded and they are not called again.

Follow up question: What happened to those who were unfit to serve due to obesity, physical weakness, etc (i.e. not disabilities/chronic conditions) who failed to shape up in training through unwillingness or inability? Were they punished, just discharged or did they literally keep going until you finally did it?