Be on-call. You need the sysadmins to be available to fix issues when they happen, even if the fixing itself takes little time.
> ... university lab ...
I developed a few web sites for universities, and they were hosted by the university. You really have to hope nothing breaks on Friday afternoon, or you have to wait until Monday morning to get it fixed...
In terms of your general point: it really depends on your business. In my last job there were 4 DBAs out of 12 total IT staff and they were constantly busy. In my current job there are no DBAs in a much larger team and yet no requirement to need one either. The two businesses produce vastly different products.
Buying and provisioning disks before space runs out is a small part of a DBA's job, and a modest constant-size task compared to predicting how fast space is running out.
You don’t remember correctly. There are multiple and to my knowledge there hasn’t been just one in at least the past ten years.