Ain't gonna lie, I laughed.
As for my headcanon.. it depends on what I'm dealing with:
If it's my own writing (where I justify multiples of the same class/character as clones), I imagine they are brought into existence to fulfil a role, and once that role is no longer needed to be filled post-MvM... well, it can go either way.
If it's a light hearted take, they get situated as far apart from each other as possible in the real world, told not to take any high-profile jobs lest a civilian recognise them and to live out the rest of their years in peace. Given they're mercs, there are a lot of issues adjusting and so they vent their frustrations in various ways. Like the odd demo giving up on a regular job and travelling the world to study (read: hunt) mysterious creatures, or another who bands with a pyro to enter the pyrotechnics business with
sparkling assassinations inexplicable fatal firework accident on the side.
If it's a dark take, they are set to be destroyed behind closed doors. Some of them might escape. Getting really horrible, there are times I imagine it like that lovely film, Moon - where their bodies break down after x amount of years and they're replaced with a fresh clone.
When I think about the game, and how Valve could possibly end the story without breaking the eternal war aspect, I imagine them winning the robot (hat?) war and disbanding... yet failing miserably to integrate into society so they come back for a reunion and decide to duel it out like in the good old days, for the rest of their lives. Like what Tidbits said.