Private Storage can be used through the Bank Teller NPC. You can either double click on the NPC with left button or click on NPC with right button, right click will show the menu and click on Use Storage to open up a private storage.

1st storage can be used for free, however, to open up a 2nd storage will cost you 10 Luna and 1 Lune to open up a 3rd storage. It costs 100 Cron to retrieve an item from the 1st and 2nd storage, but to retrieve an item from the 3rd storage costs 300 Cron. To place an item into a storage, drag the item from the bag into the storage and to retrieve, drag the item from the storage into the bag.