Just to clarify, you don't apply for an apprenticeship with the Armed Forces. There is one route into the military, as a full serving member, and you can gain an apprenticeship by successful completion of training.
Basically, it doesn't matter what you can cope with in civilian life; medical standards for the Armed Forces are designed from research and risk assessment for a very different lifestyle.
You can appeal if you can.provide information, with evidence from a professional, that is new or in addition to what is in your medical records. You haven't said why you failed the medical - and don't need to on here - but you need to address the specific issue(s) raised in your rejection letter.