This was very helpful. Your output...
open("/etc/portage/repos.conf/", O_RDONLY|O_NONBLOCK|O_LARGEFILE|O_DIRECTORY|O_CLOEXEC) = 3
open("/etc/portage/repos.conf/", O_RDONLY|O_NONBLOCK|O_DIRECTORY|O_CLOEXEC) = -1 EACCES (Permission denied)
It turns out that /etc/portage/repos.conf/ is owned by root:root. Presumably, eix is using the portage user, and in combination with my default umask=077, it did not have permission. It's not a problem for /var/git/meta-repo/ because it is owned by portage:portage. I had adjusted the permissions of /var/git/meta-repo/ and one point, but never touched /etc/portage/repos.conf/. Occasionally, umask=077 does lead to some oddities.
I have it working now.