According to this Facepunch thread, the Quake III CD comes with a 16-bit "Autorun" launcher. (Windows refuses to run it since 64-bit x86_64 does not support running 16-bit code natively.) However, you can bypass the launcher and run
DEMO32.EXE directly, which contains the actual installation program; when prompted for a file, give it
Method 1: If you MUST have a classic QIII/Q3TA installation
Instead of launching SETUP.EXE, just run DEMO32.EXE and a file open dialog should appear. Point it to QIII.DBD on the CD.
Remember to apply the 1.32 point release AFTER installing both Q3A and Q3TA. The official 1.32 point release is available from the official ID Software site - www.idsoftware.com
Method 2: Using ioQuake3
There is an engine update called ioQuake3. It's less buggy and more efficient than classic Q3 1.32
Install ioQuake3 first (http://ioquake3.org/get-it/) and then from the same page, grab the data installer as well. It will unpack your CD for you.
(KD007 on Facepunch)
In the earlier days of 32-bit Windows, many 32-bit programs would come with 16-bit installers (especially InstallShield) or at least having an initial 16-bit stage; one possible reason for this was to display a proper explanation message when the user attempted to install the program in Windows 3.x.