No, PS4 doesnt support CDs. And no, the format isnt exactly "too old to be supported", but you could make a couple of guesses as to why Sony isnt supporting CDs. One being that they tried to push their own music streaming and digital sales by not even allowing you to play MP3s from USB at launch.

How can I play CDs on my PS4?

Insert the disc, and then select the content from the content area. Your PS4™ system must enable the disc playback feature over the Internet, one time only, before you can play any BDs or DVDs. After this feature is enabled, your PS4™ system doesn't have to connect to the Internet to play BDs or DVDs.

What discs are compatible with PS4?

The PlayStation 4 can play HD and 3D Blu-rays, but not 4K UHD. Both the PlayStation 4 and PlayStation 4 Pro play HD and 3D Blu-ray discs. These discs go up to 1080p, which is the standard for most devices. UHD Blu-rays.