First of all this is a hidden/exclusive rare achievement that's only possible to get between the days of January 21st-25th which is the birthday week of the studio(amplitude) that brought us the game.

+ you can change your time to the days between January 21st-25th it will still count it as endless day and you can get the achievement

With every faction you can get the achievement!

So around 30-40 turns a random quest will appear in that the game will ask you to destroy a tiny fleet of 3 small ships and 1 medium ship (it doesn't take much effort to kill them).
When you will destroy it the quest will go into chapter 2 , the quest now will ask you to settle a number of dust halo anomalies.
When you finally settled all the required dust halo anomalies the quest will take 1 turn to process it and will give you a short answer that explains the endless day and a nice endless hero with it the achievement .