Sometimes, when Varnish talks to it's backend things don't work out. The operating system, which handles the actual writing, fails and throws an error on Varnish. Varnish will then display a 503 page or, if you are using saint mode serving stale content.
In the shmlog Varnish will log the return value of the read or write call that failed.
414 FetchError c backend write error: 24
This is just a number, an error code specific to the architecure you're running on. If you want to know more about the error code without turning to google you can use this IBM shell script.
perbu@perbu:~/Downloads$ errno 24
Errno lookup on Ubuntu 10.10:
EMFILE 24 /* Too many open files */
Not an actual solution but it might take you a bit closer to solving your issues.
Photo is by proimos and used under a CC licence. It can be found here.