There is no such thing as "unlimited" data and storage. It is a marketing gimmick used to attract people who simply don't know any better. Think about it...if you use 20 TB of disk space for your site, and the host's server has say 4 TB of space available, do you think the host is going to spend hundreds of dollars on additional drives just for you, when you are paying $5-$10/mo.? Not going to happen.
You need to come up with realistic usage for your site, and then seek a host that meets that usage.