There are three types of storage options: Character Inventory Bags, Pran Bags, and the Bank.
Character Inventory Bags
In the image above, the numbers are:
Trash Can to delete items
Gold in Inventory
Item Slots in the selected bag
At the beginning of the game, you will have one Character Inventory Bag (Bag) that has 15 slots for items. Along the way, you will receive two quest lines to receive two additional permanent Bags for your inventory.
After you have completed the "Makin' Bacon" quest, you may speak to Guard Pizaro at the Regenshein Gate to receive your 2nd Bag. This will give you 30 slots to store items.
The 3rd inventory bag is available at level 16 from Kiki's "Boring Errand" quest in the Bitter Mushroom Farm. However, you must first complete Major Krauss' 1st quest before Kiki's quest will become available.
Note: During the Boring Errand quests, you must not choose to return to Kiki at any point during the series or else you must wait several more levels before you can obtain this bag. Therefore, select the "Speak to Sister Leah" option instead of "Return to Kiki" when given those two choices.
As you can see from the image above, there are only four inventory bag slots along the bottom row. Therefore, you can only purchase one additional inventory bag. If you purchase more than one, it will not be used and will be a waste of DPoints. You can, however, purchase an additional Pran bag or extra Bank slots.
Pran are created with one inventory bag automatically that holds 20 items. You will not receive any quests for additional Pran bags. You may purchase one additional permanent Pran bag, per Pran, from the Cash Shop.
Additional Pran bags will also hold 20 items.
You may use Pran bags to store anything that is tradeable or account bound.
The Bank may be accessed through NPCs located in all of the major town in Lakia, Traband, Karena, and Disceroa.
You are automatically given a bank safe that contains 20 item slots. Additional bank safes must be purchased from the Cash Shop. A total of 4 bank safe slots are available to be filled.
Credit: Info from https://aikaonlinena.fandom.com/wiki/Inventory