I typically suggest to completely avoid "unlimited" hosts.
Like HostLumina said, there is no such thing as unlimited. Hosts that offer this type of plan, have no good way of keeping up how much resources a can be used up on their servers. Next thing you know, one user uploads a ton of data on their "unlimited" plan, on a server that is already over crowded, and all users face issues!
A quality hosts will offer set limits, and not oversell their servers. i.e. A server with 4TB of space, will hold 400 x 10GB accounts. Even if the average user only uses 1GB of space, they do not put more account on the server.
Unlimited hosts work quite differently. They tell everyone they have unlimited space, but add all sorts of other limits on what you can host, single file size, CPU/memory usage, etc, etc, etc.
Now don't get me wrong. I have seen some quality hosts that offer a unlimited plan. Not all are created equal, but be careful. Read the fine print, ask questions, and to answer No, there is no such thing as truly unlimited!