In more recent years, there were transport bags for some sizes of iMacs and the classic Mac Pros. Nothing for the Color Classic of course because that particular shape is extremely uncommon, in the sense that literally only Apple ever built a machine with those particular proportions. (Compare to say the LC520-580 where some PC makers were also building 14-inch color AIOs with enough space for a CD drive, so there's a little more likelihood there that there's a carrying case for those machines.)
Though, by the time the CC existed, Apple also sold a PowerBook for not a lot more money and if you needed "frequent air travel" portability Apple's solution was definitely a laptop, not a bag for a desktop computer.
The CC is portable as in "you can take it to the library" -- not airplane portable, really.
If there was literally any way you could avoid flying with a CC, I'd do that. Even to the point of buying or borrowing a vintage Mac at your destination and just, IDK, boot from an external disk or MO/zip/EZ135/Bernoulli/whatever configured how you like it.
My next best bet would be to check it in your luggage, just start putting clothes in a bag, then put it in, then continue around it. The trouble with that is that checked luggage is basically to be presumed it'll be handled roughly.
And, a CC on its own may be bigger than the carry-on bag size on most airlines. If not, you could probably very gingerly transport it around in basically the biggest generic padded tote or backpack you can find.
Good luck!