i think honestly he's just a lawyer. he knows the law (and he hasn't been shown to be some kind of genius who just remembers law) and chooses to employ it to diffuse any hassles  he comes across plus he's in that support group bc of the court but no one there seems to be treated as criminals? on top of that he's always dressed in expensive suits while going to work and even his working hours are from the morning and seem fixed so it makes it unlikely he's in any illegal business (like an assassin or smth). 

its also very unlikely he's ever killed given the way he was 'practicing' and all (no experienced killer would do smth like that in the stairway of their apartment esp someone who's apparently evaded law for such a long time).
he may be a coroner too, like many ppl have pointed out, but i think as a criminal lawyer - and he's been shown to be somewhat of a perfectionist - he probably knows a couple of things about murder and corpses (he also doesn't seem to be good at anatomy, just has a brief understanding of death).
it just doesn't make sense why he'd hire her, so idk he might be a spy for the government or smth. or a north korean spy even. so it'd make sense why he'd need to pass as a family man to fly under the government's radar and why he says he doesn't need to communicate much at his job. 

one last thing thou, this isn't a work of mystery but a romcom and the first two eps have almost entirely been told from her pov so i think the purpose of keeping his work a mystery is 

1. a way to emphasis on the mystery shrouding this man and his character

2. as a conflict to give the sml an excuse to butt into their lives despite knowing they're married

3. it might be a set up for a comedic reveal in the next 4 eps like for eg if the sml somehow wins her trust and ropes her into discovering his 'job', and the whole thing's v tense bc he's convinced this dude's a psychopath and she wants to prove her crush's innocence but then they find out he's just a regular office worker who offends a lot of ppl w his blunt words and brusque mannerism it would be. hilarious actually.

idk what ever it is but i really do think that the only reason he himself hasn't mentioned his job is either bc he's a spy or bc he's genuinely that dense and blunt and doesn't see why he should divulge too much information about himself in a fake marriage. i think maybe if she were to press a little he might have just told her.

I think Prosecutor is probably it too BUT I'm pitching for him being a crime scene consultant, like a pathologist/haematologist/forensic scientist.  Blood spatter patterns.  Matching weapons to injuries.  All those things requiring clinical disassociation...

The biggest clue so far was his line about, ".My job is to end people's stories." (Sorry if that's a misquote!)

he seems like a profiler 

I don't think he is a prosecutor or lawyer because he said his job isn't very common

But it's definitely smt in that area. When he was eating outside he had id tucked in pocket and it was blue rope (dont know how that thing that you hold your ID is called). And in most dramas policers, prosecutors have it in blue :)

• Korose, it's a lanyard.  I only know because a work colleague used the word all the time and I had to look it up