I've installed Debian Squeeze (note: not Wheezy) on a machine with a 6 GB HDD and 256 MB RAM. Worked fine. The Squeeze base system + X + openbox took about 1 GB.
You can probably get a smaller Slackware install, but you have to know which packages you need. You won't be alerted if you lack something crucial -- you are recommended to install all of the Slackware software series in a normal install and the dependencies in the base system are, as far as I know, not even documented.
In Debian you install the base system and then apt-get install
what you need. It's much easier. I recommend putting this in /etc/apt/apt.conf
before installing anything to pull in fewer useless dependencies:
If some package doesn't work, check the recommended ones.
150MB to launch a modern web browser (with a blank start page).
Or 3 MB to run Lynx. About 9 MB for Links with X graphics