At least you'd need to turn off base URL checking then because Magento would not match "//example.com/" to the requested URL, leading to an infinite redirect loop.
But if you want to use HTTPS everywhere, using https://example.com/ as unsecure base URL will work.
Update: I use protocol relative URLs for "skin" on a EE 1.14.2 project now to always load web fonts from the same protocol (otherwise there are cross origin issues). It works fine, but it was not possible to enter the URL in the Magento backend, so I had to update
core_config_data directly in the database. Saving the configuration in the "Web" section results in a validation error because Magento expects any URL to start with
http (or with the placeholders for un/secure base url)