Web hosting with huge storage (around 5 TB)

Currently, I am working in a web project, where the aim is to develop a web application with a huge user visits, which are going to upload videos within this App. I have been searching for a while to find a way to maintain such a huge storage capacity. What I have seen so far, all hosting providers offer "unlimited" storage capacity, but actually this is not unlimited and does not fit to my requirements.

My primary idea was to find a way to relate my host with a cloud, or even to use a cloud directly as a host (if something like this is possible). For example, host the wen in normal host and at the same time build an API which will manage the videos. The purpose of the API will be to upload and download the videos from cloud, making them accessible from the host directly.

Thank you in advance and I would really appreciate any suggestions.