Hardware was never an issue - my 1G internal hard disk is only about 20-25% full, and the Mac Mini itself is on the list of Macs supported by Catalina.
I did finally and accidentally discover the cause of the problem. I was using the Firefox browser to access the App store and apparently it has a problem with the Apple Store using iTunes. Someone else thought it might be a userid issue, so I logged on as Administrator, and on that userid, the only browser I had ready access to was Safari. When I tried using Safari, I was able to successfully download the Catalina installer. So, out of curiosity, I logged back in as my regular userid, and attempted to access the Catalina page using Safari - and was likewise successful. I'm running the installer now.
So: long story short: browser issue.