The game needs a warehouse type of building. Or at least the option to upgrade the stockpiles. You can solve the problem in several ways by now.
Guide to Increase Stockpiles
The “Warehouse” in this game is the Market you start with. By adding more Market Stalls you can increase stockpiles.
The only thing that is a little annoying is that each inventory place in the Market Stall seems to be set to a specific resource (Market Stall 1 can only hold one stack of Fish, Cider, Bread and Simple Cloths) and you cannot assign them individually to make like one big Fish stockpile.
Build a Foundation
You can just build a foundation and put some crates (construction tab), select what you want to store. Or use harbour as storage (add as many crates as you want).
There is Harbor zone in the game. You can build another Harbor with stockpiles if you like, or build stockpiles from the Construction menu anywhere on the island, without a zone.