Endless Day is an annual event that is built into the game that runs from January 21st to January 25th to celebreate the anniversary of Amplitude Studios. The event first appeared on their 4th anniversary, in 2015. You can unlock the hero Esseb Tarosh and the Steam achievement "Endless Day" while the event is active.
To unlock the archivist, Esseb Tarosh, you need to start a new game of Dungeon of the Endless while the event is active. The hero will be available in your pool right away, for the duration of the event. Like any other hero, you need to survive at least three floors with him or finish the game with him in your group. If you manage to do so, once the event ends, you will be able to keep the Esseb Tarosh in your pool of heroes.
To unlock the Steam achievement “Endless Day” you need to start a new game in any difficulty, hire Esseb Tarosh and complete the game. You must start and complete a game between January 21st and January 25th in order to get the achievement.
The event triggering is based on your computer clock, so you can change the date on your operating system in order to trigger it manually.