You can upgrade your barn, which expands its capacity.
1. Tap 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. Tap the Increase Storage to... button.

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