From Fallen London Wiki
These are stories that come to life as time passes. Upon receiving the "A living story will begin soon" message, players must wait any time from several hours to several days to continue their Living Story. They will receive an in-game notification under their Messages tab once the next part of their story begins. The first living story that players will encounter is Time, the Healer, though a number of other stories exist as well. These are used to time-gate players, either because the rewards are high, or the story calls for a break, perhaps for suspense.