Is this an application for students and you are a student on the campus? And is the funding from the school or something else? If you can tie it into the school, perhaps you could host the server on school resources in their datacenter. At my school, the campus Linux mirror was actually run by a student group, and just colo'd in the IT datacenter.
If this is something that you're developing on your own and you're trying to build traction, you could also host it on your own computer on your internet connection. For as much as anyone needs it to be up until you're making money, that's fine.
You could start off with a cheap VPS or the free tier on heroku or Amazon aws and see how it goes.