Why do I keep getting 0x80070005 when updating Windows 10?

After upgrading to Windows 10 from Windows 8.1, I keep getting error 0x80070005 whenever I try to apply an update or add a Windows feature, which is E_ACCESSDENIED.

When I try to monitor with Process Monitor, I see FSCTL_SET_EXTERNAL_BACKING on some files in \Windows\Servicing\Sessions\*.xml is failing with STATUS_ACCESS_DENIED, which is confusing. Messing with permissions doesn't help.

What is the problem?

The hint is logged in \Windows\Servicing\Sessions\CBS.log:

Info  CBS  Failed WOF compressing file \\?\C:\WINDOWS\Servicing\Sessions\{name}.xml

WOF refers to the Windows Overlay File System, used for file compression in Windows 10.

Turning CompactOS off (via runnin Compact /CompactOS:Never in the Command Prompt) makes the error go away. (It's not clear to me why the problem even exists, though.)