GameCenter does not currently have support for downloadable content. You could peer-to-peer share small payloads of data between players using the matching API, but that's pretty clearly not what you want.
In thinking about other services, the first/easiest thing that comes to mind for me would be to use Amazon S3. It's super-simple, reasonably cheap, has good content distribution, availability, etc. (Sure beats running your own server, anyway.)