So it seems... Even https://cloudflare.com/ itself is down
The API wasn't working perfectly, but with some retries we were able to change the config for our domains.
Seems to be back up and running for me.
But in all frankness, if Cloudflare's own site did not run off of Cloudflare's infrastructure, why would anybody trust them with their websites?
In case of network issues affecting their proxy, being able to change your configuration to allow direct traffic would be really nice.