On-site storage would be the files you use to run your website, such as pictures, CGI or PHP files, maybe a database for users and passwords, etc. Off-site would be files not essential to your web site such as hard drive backups, packages for sharing or storage, a repo with multi user access and downloading, etc.
Most web hosts would shut you down or bang you pretty hard with band-width and usage fees. For what you have in mind, you should contact them first and make sure it's ok so you're not out a bunch of time and money.
Some of these hosting companies promise you a grand piano but deliver a swollen organ, if you get my drift.