I'm making a wild guess that you are looking at non-printing characters. Look for an icon on your toolbars that looks similar to a back-to-front capital P and click it. See if they go away.
Been looking around for a good picture to give you a link to but I can't find any that show the toolbar button. This is the symbol I mean (if it shows properly...):
It's normally on the Standard toolbar, just before the drop-down arrow that lets you choose if you have the page at 100% or show two pages at once, etc. But it's looking like that might not have been standard in Word 2003.
If you can't see it, go to the Tools menu and then Options. On the View tab, look for formatting marks and make sure all the options are unticked.