Cloud computing is not simply for storing your personal files. It is a solution for managing the data center for your business. I can see it has attraction for small and middle size of businesses. By using the service, you can eliminate the headache of keeping your data center up time, backing up your data etc. And you can access your data and application even when you are on the airplane. Obviously, you cannot carry your data center with you when you are traveling.
Without internet, you can do nothing if your application is web enabled. It doesn't matter if you are managing your own data or using cloud computing service.