I built http://jsonblob.com for this purpose. It has an HTTP API as well as we nice GUI interface to edit your JSON. It's open sourced and really just a thin webapp on top of mongo, so it's easy to run your own.
Yeah, it uses that for the editor, but also has a full HTTP API (http://jsonblob.com/api) behind it for storing the JSON and accessing it from outside the browser.
Could be used for notepad API or something public. That said, it seems that services like Parse and Firebox seem more robust and also have faster response times.
I can't see the use case of this. If it was editable after saving then that would make sense, but this is just static. And being static, you're better off just bundling it with the app. Unless I'm missing something?
Nice presentation though, the site looks nice.
EDIT: I see you can create and update through the HTTP API, which makes it much more useful.
Why not use https://www.firebase.com? Seems more reliable (Privacy Policy, ToS, SLA) and easier to start using, and it has a free tier if you need something quick for development.