On screen: An error occurred while updating <game> (disk write error)
This error is produced locally. It does not appear because the service is down, so the fixes below are worth working through in order.
Steam could not write or rename a file inside steamapps/downloading or steamapps/common. It is a local write failure, not a download problem.
The usual causes are NTFS permissions on the library folder or a read-only attribute, a stuck zero-byte or locked file left behind by an interrupted download, antivirus or Windows Controlled Folder Access blocking Steam.exe from writing, a full disk, bad sectors, a failing SSD, or a library sitting on exFAT or a network share.
This is almost always local. The server-side equivalent looks different — Steam says content servers unreachable when the problem is on their end.
Operator documentation: help.steampowered.com. Valve's page covers update and installation issues generally: “The error results when Steam is not able to complete the installation. The specific reason for the error will be shown in the parenthesis at the end of the error message.” The older widely-shared support.steampowered.com link now redirects here.
Steam could not write or rename a file inside steamapps/downloading or steamapps/common. It is a local write failure, not a download problem.
This error is produced locally. It does not appear because the service is down, so the fixes below are worth working through in order.
Start with: Open Steam's own content log. In your Steam folder, open logs\\content_log.txt and search for “failed”. It names the exact file and the operating system error code. No guide on the first page of results tells you to do this, and it turns a guessing game into a two-minute fix.