Very recently I acqired a Toshiba T4700CS laptop. For those of you who aren't familiar with it, here are the specs:
What I'm looking for is a fast, easy-to-use and very small install of Linux that will let me browse the internet (with decent graphics, so Lynx is out ), check e-mail, exchange IMs, and word process without being extraordinarily slow and (if at all possible) not eating up more than 50 MB of space.

I've used Linux in the past, but had bad experiences with it - I can get around and do some basic file work with competency. (LFS seems daunting, and I'm hoping I won't have to resort to that - I might just decide to leave Win95 on in that case.) I've tried installing muLinux, but it boots up EXTREMELY slowly, taking longer than Win95 - it takes several seconds to get through each disabled package. (This is running it from a restart-in-ms-dos-mode)

Currently I've got Win95 with MSOffice on the 'top, but have no way to back it up as the Microsoft Network won't work; FTP does, however, but the dos client won't auto-recurse into directories - a decent backup tool would be mucho appreciated as well.

Hope I'm not asking too much.

You can look at DamnSmallLinux, PeanutLinux or TinyLinux... or Slackware.... but 200MB seems a bit small for Slackware.

mulinux uses a small c lib instead of glibc, the c lib it uses sacrifices speed if it gains less space, mainly only good for embedded devices that have small storage areas

i don't think you want to do LFS (compiling needs a lot of ram, and hard drive space for sources... yours just wont cut it.... play with a 486 it could take days to compile glibc (it takes my 586 laptop about half a day to compile glibc ... )

so get peanut or damnsmall linux, as installing from source codes on your machines just wont work that well (you can compile small packages, in a few minutes ... would take a new computer seconds) . but nothing big)

DSL seems to need a CD; PeanutLinux's site either isn't in English, or it needs 999MB or so. Or I've got the wrong site altogether. :-/

TinyLinux looks like it might be good; Slackware looks like it *could* work, depending on its size.

Anyway, here're the two sites I've found for PeanutLinux:
http://www.gudanglinux.or.id/content1peanut.html
http://www.ibiblio.org/peanut/

Am I missing one...?

DeLi linux is designed to run on 486's