Hetzner just answered us, saying the problem is resolved.

An error in the distributed storage system caused a temporary loss in performance. The error has been resolved and the storage system is fully operational again. 
Please excuse any inconvenience this has caused.

Everything on our end looks good.

The server regained access to the network storage. Every service is back online. We are monitoring the issue.

We are investigating a storage issue on our web-01 server. It seems that the Ceph network storage used by Hetzner crashed. Our internal services are now unavailable, as same as our website. No customers services are impacted. We'll keep you informed.