Running Linux on a read only disk - possible?

Yes, it's possible to run a Linux server with read-only media. See Devil Linux; those servers run off a CD disc.

You can create a writable space for things like log files by using a RAM disk. Space will be very tight, though. I wouldn't recommend doing this except for very specialized circumstances.

In most cases you can hybridize the process; most of the server is read-only but there's a scratch space for log files with an internal drive or USB drive.

You can also do this with bootable Linux routers, as I've seen them made to boot from a floppy that's read-only.

Another drawback is updates. The thinking was that even if someone were to crack some flaw, they couldn't do much damage since they can't alter system files or anything like that, I guess. Periodically you update the master image or get the newest ISO image and use that to boot from as the update; with the Devil Linux server I had experimented with I had the ISO image to boot from then used a USB disk to pull the configuration-specific files for that server. Alternatively you can edit the files required for your particular setup and re-integrate them with the CD's /etc directory and re-burn the disc.

Anyway, yes, it can be done, but it's a maintenance hassle and something to only be considered in very specialized cases.