You can upgrade your barn, which expands its capacity.
1. Choose Upgrade at the bottom of the Barn screen.
2. Collect the required building materials (hammers, nails, and paint) or buy them using Township Cash.
3. Select the Increase Storage to... button.

You can also sell unwanted items directly from your barn to make room for new ones.
1. Select the item you want to sell.
2. Specify an amount.
3. Select Sell.