I remember around 2002 running my home router without any hdd on fli4l - a single floppy linux router distribution. I slept in the same room as the router was, hence I wanted a solution without a noisy hdd from that era.
There was also tomsrtbt[1], which was a staple in my "rescue" floppy collection. Along with the QNX floppy[2], which came in super handy when using a cyber café or a friend's PC, and you wanted to avoid all the keyloggers and malware.
Trinux was another such utility. It spanned multiple floppies, but booted to RAM and could effectively run from one. The additional disks were for additional tools/utilities.
Still apparently available on Sourceforge.net, though last updated in 2013:
At the same time, I was running a home router without any HDD on LRP, Linux Router Project, which was a distribution from Swansea Linux, and was a floppy image, that decompressed into RAM, and then chrood to the RAM image. Really nifty, except for the 486 machine had a Pentium Overdrive, which was vulnerable to F00F, and we got owned... only to reboot again, and back to our normal image.
Since it had no hard disk, and no monitor, it was quiet, and used little power.
I remember some people booting firewalls off CD-R's. Burn a new one when it needed an update. Booting with no writable media at all, truly an immutable distro.
[1] https://en.wikipedia.org/wiki/Tomsrtbt
[2] http://toastytech.com/guis/qnxdemo.html