On linux, what does it mean when a directory has size 0 instead of 4096?

I'm guessing youre using autofs for your home directories.

A couple of different pseudo-filesystems will show up like this. Pseudo-filesystems being filesystems that dont really hold files. You'll notice /proc is also like this. Basically its because there is nothing in /home, no files, no anything. When you try to access it, autofs see's the request and tries to mount up the appropriate home directory as requested. With proc, those arent really files, theyre just representation of files as provided by the various drivers (and kernel) running on the system.

Edit: Shut off autofs and it'll show up like a normal directory.

df -i might be interesting - could you be out of inodes on that filesystem?