We still use tapes at my enterprise. I'm on the network team, so I don't really work with the tapes directly, but it doesn't seem as inconvenient as it might sound.

We have a few large "tape libraries" in our data centers. The tape libraries have a robotic arm that handles the bulk of the tape swapping and organization. Humans are involved in loading the tape library and carrying the tapes offsite for storage.

I'm speaking as someone who has had a couple of casual conversations about the tape libraries and am not involved in the management of them in any way. They could actually be a huge pain.