Well, three things pop into mind while reading this.

• SSH has been implemented in pure perl. You could do that. Your limitation is based on size, not dependency, so any module functionality you reproduce will still take up (possibly more or less) space on your system. Is an SSH client binary smaller than a gzipped ssh client script including the pure perl implementation?
• You could think about all the things you (or others) used to do with shell scripts before they had perl. Ok, now think about how much you could improve the functionality of your code when you discovered sed, awk, or grep. Now rewrite it in perl.
• Building on the previous item, if you had core modules like Socket you could build all kinds of marvelous tools. A web browser with simple markup (lynx), a file server that takes a filename and dumps contents back down the pipe (webserver), a multiuser chatsystem (ychat), a lush descriptive text based game (mud/bbs). I've seen all these implemented as shell scripts, they're cake in perl. They are all very simple protocols you could emulate with a telnet client, could you imagine trying to do something like this on a heavy modern protocol based on XML wrapped packets, bleh.

You could spend time recoding your second program, the one you wrote right after Hello World:

10 print "elwarren is cool!"
20 goto 10
[download]

:-) The question is kind of like, what should I code next? It's all up to you and your imagination.