If your map is full of storages, I imagine that most of your bots are set to "Repeat Forever", including those that produce resources (cutting down trees, mining ore, etc.) and those that put the resources into storages. In that case, even if there were bigger storages, those too will eventually get filled and the problem remains the same.
You might find as an enjoyable challenge trying to regulate the bots that produce resources so that they stop doing so when the storage is full and later start producing those resources when the storage is not full. Try also using the "Shout" and the "Repeat Until Hear …" commands so that bots that identify that a type of storage is full or not full can control the bots that produce the resources and fill the storages.
Of course, the folk need to be fed which should use up the stored resources.