You could use the Dropbox API with oauth. The same files in Dropbox can be used to sync the web API, phone apps, and desktop apps.

You could also roll your own solution with S3 or something like that. The point is that you will be using fairly common and popular services which aren't likely to go away anytime soon, and aren't operated and maintained by yourself.

You could also take the approach 1Password does: operate your own sync service, but allow users to use Dropbox or iCloud instead. The onus is on them to make sure the files are in the right places, but they effectively have total control over the files, which a lot of people like.

An alternative to look at is the Couchbase Mobile stack. The stack in includes the Couchbase Lite database (NoSQL), Sync Gateway, which enables syncing and more, and Couchbase Server. All open source. It's pretty easy to run all yourself, or on something like AWS. The db runs on many platforms, both mobile and desktop.

Disclaimer: I work for Couchbase.