On screen: An error occurred while updating [game] (content servers unreachable)
This one genuinely goes both ways, so the first job is telling which you are looking at.
Your client cannot reach the content delivery servers for the download region it is set to. The region is a setting, not a fact about your location, and that is the key to diagnosing it.
A single region can be degraded while the rest of Steam works perfectly — the store loads, friends work, games launch, and downloads fail.
Change your download region and retry. If the download starts, the problem was that region and it is Valve's. If it still fails in several regions and other downloads fail too, it is your network. Valve's own article on server problems recommends the region change, so this is their suggestion as well as ours.
Operator documentation: help.steampowered.com. Valve does not document this error specifically. Their only relevant article covers update and installation issues generally, and says: the error results when Steam is not able to complete the installation, and the specific reason will be shown in the parenthesis at the end of the error message. They confirm the format and never explain the reasons. So anyone quoting an official Valve definition for the text in those brackets is quoting something that does not exist.
Your client cannot reach the content delivery servers for the download region it is set to. The region is a setting, not a fact about your location, and that is the key to diagnosing it.
This one genuinely goes both ways, so the first job is telling which you are looking at.
Start with: Change the download region. Steam, Settings, Downloads, Download Region. Pick somewhere else in your country or a neighbouring one, then retry. This is the fix in most cases and it takes seconds.