I only remember some elements and only hazily - nothing of the ending. An internet link took me to the story, I read it online, but it wasn't dated so I don't know which era it's from. I, however, read it about 2 - 3 years ago.
The story itself starts in modern-ish times (might be during or around the time of a big war/conflict)
The main character is a man who doesn't realize he's immortal and finds a partner he loves (I think moves abroad with them). As the partner ages and he doesn't they realize something is up. Then the partner dies (old age or cancer, can't recall) and our main character falls apart, living in seclusion with his grief for multiple years.
Fast forward (in my memory) to him becoming a mercenary to heading his own mercenary team, to getting out of the mercenary game.
At some point humanity creates a computer or artificial intelligence of some sort that takes care of them, the only other ageless consciousness.