I was thinking in regard to a webservice, you could write a webservice that was unhackable with enough time and effort, although as mike point's out, it does not stop someone from writing a fake website and stealing your users passwords, or a rouge sys admin from modifying the service.
While not a technically a hack:
A great example of both a great scam to get money from people and also a service that you don't have access to the encryption keys would be the Cryptolocker virus.
The files on your system are encrypted, they keys do not exist on your computer, unless you give the friendly folks who have sent you the email 3 bitcoins, you are unable to unencrypt the files.