Well, they USED to make computers that were only word processors. You can get pretty close by doing the following...
1) Get a computer that has an absolute MINIMUM of add-ons. If you have a knowledgeable friend, you can probably even build one from spare parts that people (and companies) are throwing away. Heck, start talking to companies and you might even get one of them to PAY you to dispose of their old computer equipment. I started doing this as a sort of charity, to build computers for kids in low-income families who could use them for their school work.
2) You don't need a hard drive. You may want one for convenience, but you won't really need it for anything except storing your documents, so it can be REALLY small.
3) Get yourself a copy of Ubuntu Linux. It's free. You can either get Ubuntu by downloading and burning the CD, or they will even MAIL you the CD for free...
http://ubuntu.com
Ubuntu (and most other versions of Linux) comes pre-installed with OpenOffice, which includes this word processor...
http://www.openoffice.org/product/writer.html
4) As I said, the computer doesn't necessarily need a hard drive. Unlike Windows, Ubuntu can run completely from the CD-ROM, though it will admittedly take a bit longer to boot than if it were on a hard drive. Even then, the hard drive wouldn't have to be too large... 3 or 4 gigabytes is enough.
You can also store your files on a memory stick (jump drive), which also makes you portable in case you want to have a look at your documents on someone else's computer. And yes, OpenOffice Writer can read and save Microsoft Word files too.