Yes, you can. The beauty of ebay is thatyou can sell almost anything on there - a piece of gum chewed by a celebrity, a car, jewelry, anything. Storage units are valuable and you could get good money for it.
You can auction off almost anything you want on ebay, including a storage unit. There might be difficulties in the arrangements and it might not be popular because someone would have to be near by and storage units are not a popular search item on ebay. This would result in a potentially lower auction price than you might want.
No you don't own the storage unit you just rent it. You can only auction off something that you own. Unless you are the owner of the storage unit company.
Large items on ebay need to be picked up by the customer for the most part. If the buyer and seller can reach an agreement, it can be dirven half-way or a similar agreement. The problem with shipping is that it is too expensive and this could cost just as much or more than the item being sold.