Agree that getting it off the ground is more important than under cover, in most cases. Some equipment will get rain down into crevices and gears/chains, but all of them will rust from the bottom up if left in the dirt or grass. Setting them up on pallets, tires, logs, railroad ties, landscape timber, etc will be a great start.
Carport or loafing shed both work, it depends what fits in your space and access better. Make sure it is tall enough to lift your loader most of the way up—high enough to lift equipment with a chain from the bucket.