I got this error on a cPanel hosting account, where the disk quota (for the account) was not an issue. After delving into it for a while, I found that cPanel's "Select PHP Version" tool was set to v7.1 but multiple WordPress diagnosis tools were detecting v5.6 was actually running, and they also detected file system issues.
The File Manager tool in cPanel showed all the permissions were correct, and the folders were writeable.
One of the diagnosis tools I used was part of the Wordfence plugin. It was available on the Dashboard menu at Wordfence > Tools > Diagnostics (tab).
I reported the issue to the hosting company, and it appeared to fix itself overnight (the person who responded to my support ticket indicated they didn't fix anything). I think it's likely it was a symptom of a problem that affected multiple hosting accounts, and somebody else within the hosting company probably detected it and fixed it.
In case others encounter a similar issue, I hope this answer helps them spend less time trying to track it down. Once I discovered the incorrect PHP version appeared to be running, I figured it was probably not something I could fix with only cPanel access.