I remember reading a story online years ago, but I can't find it anymore. The idea behind this story (pretty sure it was a short story, not even a novella) was some kind of many-worlds thing: there are an infinite number of universes, and all manner of coincidences happen somewhere, and you never observe the universes in which you happened to die. So, your subjective experience is that through a series of increasingly unlikely coincidences, you somehow manage to stay alive. In the end, the protagonist outlives everyone he knows, and eventually all other life on Earth; I remember the story ending when some aliens discover Earth and start investigating him, but I could be wrong about that bit.
Probably one of the stories mentioned in the Wikipedia article on Quantum suicide and quantum immortality in fiction. I think the one you're looking for is "Divided by Infinity" by Robert Charles Wilson; you can read it here