I have a very old AirPort Extreme, the A1408. Is it possible to install Linux on it, using the AirPort functionally as a hard disk, and then boot from that? I have also heard that AirPorts run NetBSD. Can you boot into that and run commands?
You're right that your A1408 (2010 3 spatial stream 802.11n) AirPort Extreme runs NetBSD, but Apple has never exposed a way to SSH or telnet in, nor have they exposed a serial console to do the same thing locally. Apple's firmware update process is pretty secure, with encrypted connections and firmware image signature verification, which adds layers of complexity that make it a much less interesting target for hobbyists. Jailbreaking AirPort Extremes just isn't as enticing as jailbreaking iPhones.
No, you cannot install Linux on an AirPort Extreme. They do not have the hardware needed to run an OS, just some firmware to run networking functionality.