Storage boat showed up. I don't want to store anything and can't move on in game. What do I do?

Sounds like you may be in the middle of the tutorial. Just follow the instructions, then afterwards, you can put things they way you like. Storage is unlimited and very helpful when you want to move many pieces to different areas of your island. Only caveat is that you lose the Flower value of the pieces while in storage, but they will return when you place them back on the island.