Bag space gets a lot better as you level up. In the early game, space will be at a premium and you'll always feel cramped for space but when you reach the city (and the broker) and as you level up and earn some money, you'll find there is a lot more space available to buy. As noted above, bags are available for purchase or through questing that are much larger and there are also strong boxes available that are cavernous in size! (btw - tailors make bags and carpenters make boxes)
Also, have you been to the bank? There is a bank on the island and in it, your character has 20 more slots of bank space (12 personal and 8 shared*) each of which will hold any size bag or box available in game. If you were to populate each bank space with a 48 slot box, you would have almost a thousand slots for storage!
note:
*The 12 personal slots are unique to each character you create and the 8 shared slots are common among all your characters.