What Happens if Hive Web Shuts Down?
You could most certainly spin up your own server :) Included in the Github repo is a Vagrantfile that uses Ansible to deploy and provision an instance of hive-js. This can be handy not only for creating a local test instance, but also for deploying to more public environments such as DigitalOcean or AWS.

Alternatively, because Hive Web implements BIP32/BIP39, other wallets implementing the same standard should allow for compatible 12-word seed phrases to be moved between clients. In addition to being a high priority item for Hive OSX, it seems that most if not all Bitcoin wallet developers are striving to implement these same standards.

If all else fails, you could use something like http://bip32.org/ to manually recover your master extended key and sweep the funds elsewhere.