I can think of at least 2 possibilities:
(1) Your CD-ROM has degraded over time, and the data it holds has become corrupted. While this would indeed suck, unfortunately nothing lasts forever.
(2) You may be running up against a problem initiated in late 2015, when a Microsoft update disabled the
secdrv.sys driver on all current versions of Windows (Vista and up): although this update could be rolled back and then blocked from re-installing on Vista to Win8.1, this is
not possible on Win10.
That driver was used to run copy-protection/ DRM (ownership-verification/CD checks), and ever since the update, CD-based versions of many games published around 1995-2010 (including Civ3) can no longer be installed/run.
See this thread for more details.
The simplest/easiest way to get around both the above problems is also detailed in the linked thread. The TLDR version:
Re-purchase a download-version of Civ3(Complete), e.g. from GOG
or Steam (for around 5 local currency units, and less during sales). Having done so, if you really only want to play the original 16-Civ vanilla version of Civ3, you can patch your installation with a NoCD-version of the vanilla
civilization3.exe v.1.29 executable (Firaxis supplied a legitimate version, which is still
available online at the German PCGames magazine website).