Can your character kick-ass despite working in a Dry-Cleaners? Do they have strange, unexplained tattoos and gnarly scars littered across their bodies? Do innocuous things cause traumatic, immobilizing flashbacks? Where did they get those guns and how the hell do they know how to use them all?
When all of these qualities fit the profile of your character, then they may or may not have been involved in a form of military or law-enforcement services and have since moved on to other things. One does not just forget these skills and may have to use them when they are needed. The reasons they have kept their past professions a secret and why they continue to do so can be for many reasons, maybe out of a sense of guilt for their on-duty actions or because they feel like it is not worth explaining. Either way, all we can do is guess.
The invention we refer to as "the Military" is a multifaceted socio-political organization that has one purpose: search and destroy. Oh, and to serve and protect their country. And provide tuition for its members. And instill discipline into those who have none. To describe the purpose of a military is a philosophical debate unto itself, but this also has to keep in mind the various divisions and branches (Army, Navy, Air Forces, Marines, etc) and their types (artillery, infantry, special forces, etc).
This isn't just something that applies to a specific military organization either. It can work under homeland intelligence services like the Central Intelligence Agency (CIA), international agencies like the International Criminal Police Organization (INTERPOL) and any other Government Agency of Fiction that has Fun with Acronyms.
See also Retired Badass. When averted, see Phony Veteran and Wannabe Secret Agent. The law enforcement side of this trope is an opposite to Ambiguous Criminal History.
Ambiguously Trained has been suggested to play 2 roles. Click below to see other actors suggested for each role, and vote for who you think would play the role best.