Most distros should come in at under this size if you don't un-install anything superfluous. Their default package sets may not be what you want, but look into "net-install" routines. Several distros have them. For Arch it's the default, debian also has such a thing.
My distro of choice (PLD-Linux) comes in at about 80 megs for a default working install that does everything you request and isn't stripped down (i.e. the man pages and such are there), then I add on packages from there. If can easily take that down to ~30M by not including things like larger shells, man page, all language locales, etc. On a desktop system where I add a whole bunch of stuff it starts working up to a couple gigs. I have installed and run the same distro on 16M compact flash cards and multi-terabyte RAID arrays.
My point is that just because some distros install a lot of stuff doesn't mean you have to have that stuff. Any good package manager will let you install a minimal set of packages to run a system. You should consider what distro you are comfortable with administering, not which ones comes with the smallest package set by default.
I would lean towards recommending ArchLinux or Debian, but don't rule out others until you know what is involved in getting a minimal package set running on them.