It seems from a quick read of the design document [1] that the important thing here is the updating of refs using a compare-and-swap, ensuring that the synchronisation issues that typically occur with using Dropbox as a remote cannot occur. Objects are always* safe from this since the file names are SHA1 hashes.
Edit: The only downside I can see with this is that the remote will never have garbage collection take place, so unreachable objects will continue to take up space indefinitely.
[1]: https://github.com/anishathalye/git-remote-dropbox/blob/mast...