0x800705b4 is ERROR_TIMEOUT reported through Windows Update: an update operation, usually the install phase, took longer than Windows allows and was abandoned. Unlike the download timeouts, this one tends to appear during installation, or while the update check itself grinds without finishing.
net stop wuauserv, net stop cryptSvc, net stop bits, net stop msiserver. Rename C:\Windows\SoftwareDistribution to SoftwareDistribution.old and C:\Windows\System32\catroot2 to catroot2.old, then restart the four services with net start. This rebuilds the download cache and the update database from scratch.If installs consistently time out on an otherwise healthy PC, install the specific update manually from the Microsoft Update Catalog, where progress is visible and the timeout behavior differs. Check drive health too, a disk slowing down with age drags every install toward this code.
What is the difference between 0x800705b4 and 0x80072ee2? Both are timeouts. 0x80072ee2 is the network layer giving up on the servers (download and check stage). 0x800705b4 is the generic operation timeout, more often the install stage on the machine itself. Slow disks and heavy load push toward the second; network problems toward the first.
My PC is old. Is that the whole story? It can be. A mechanical hard drive near full, on a machine doing its monthly updates while antivirus scans and cloud sync run, is the classic recipe. The idle-run and disk-space steps are aimed exactly at that situation.