Aside from selling your items in the Farm Orders board, and to your friends through the Farm Stand; you can either upgrade your barn or delete some of the items to make space.
To upgrade your farm, you will need various materials which you can get from foraging, and through explorations on the many Places of Interests available.
Tip: Tapping on the required material shows you where you can get it.
To delete items in your barn, all you need to do is select the item, tap on the "X" button found on the upper right corner of the item's image, and click on the confirmation button which appears next.
The "X" button will only appear once 90% of your total Barn capacity is filled.
Note: Tapping the "X" button will remove all of the same items in your barn. Please choose carefully!