When you first access the Pokémon Storage system via Rotomi or the Pokémon Box Link, you will find eight Boxes there to store Pokémon in. If you put at least one Pokémon inside of every Box, then leave the storage system and come back, more boxes will be added. This trick can be repeated until you've unlocked the maximum of boxes: 31. Each of these 31 boxes can hold 30 Pokémon, allowing for you to store a grand total of 930 Pokémon on one save file.

Unlockable	How to unlock
Defeat Eternatus	Box 32
Deposit Pokémon into all boxes 1-16	Boxes 17-24.
Deposit Pokémon into all boxes 1-24	Boxes 25-31.
Deposit Pokémon into all boxes 1-8	Boxes 9-16.