For years, when I've found an article on the web that I wanted to keep, I've printed the page. That wastes paper, and I usually end up throwing out the stacks of printouts. Recently, I've adopted Evernote, which allows me to save the articles both on- and offline. It supports Windows and Mac, most browsers, as well as Windows Mobile smartphones and now, iPhones as well. I was a very early beta user and ended up uninstalling the software because of problems it caused with my PC, but the glitches have mostly been worked out and it's now available to everyone in open beta.
If you run the desktop version, you can store documents locally; otherwise, your documents will be kept on their remote server. (You can also sync local and remote documents so that all your devices have access to your complete library.) You can organize documents into multiple notebooks, or repositories. Documents are searchable, tags can be added, links remain live, and you can always get back to the original webpage, even if you only clipped a portion of it.
The basic version of Evernote is free (that's what I'm using), but if you find yourself using it much more heavily than I do, you can upgrade to a paid subscription with more storage space. It's worth a try.