I think you would have to buy a hosting account that allows you to sub-sell hosting space to customers. This is called "Reseller Hosting". But I am not sure if the company allows you to sell hosting space to others for them to store files and folders.
There is kind of a funny rule: most hosting companies will tell you that if you want to buy a hosting account that allows you to simply store files (for the purpose of backup or to be able to access these files online), you would need to subscribe to a dedicated hosting account.
So the questions I would have then, while researching web hosting companies:
1. With a "Reseller Account", will the hosting company allow your customers to simply store files and folders?
2. If not, can you do what you want to do with a dedicated hosting account?
3. Does the dedicated hosting account fit your budget? Some dedicated hosting accounts can be pretty pricey (range between $100 and $200/month).
I realize that this post contains a lot of info, and I am not sure how comfortable you are with the terminology, so please do not hesitate if you have any questions or need clarification!
Hope this helps!
Sherif