Gmail email offers free storage of all your emails, so use it! Sign up and send yourself all your important files but don't delete them after you've read them. They'll be kept safe by those nice folks at Google.
Most new computers come with a CD and DVD writer. These days you can just stick a rewritable DVD in and move your files over. Next time the children are over, get them to do it for you. Be warned that once in front of a computer they'll spend the next 10 hours chatting to their friends on Bebo..
If you have an old computer sat in the corner, then make use of it! Save your files on it, and when the latest virus takes out your main computer, you'll have a backup.
Now, we are not suggesting that you deliberately break the rules of the IS department at work, but many of us have spare space on a work computer that can be handy to keep files that are better not lost. Not a great place to keep that “100 Reasons why I hate my boss” epic you wrote, and not a good place for your Valentine poetry either if the nerds in the IS department like to nose around. Advantages: Costs nothing.
Remember when you bought that digital camera 5 years ago? Remember the fortune you spend on the memory card? Old camera memory cards can store anything, not just photos, so connect it to your computer and drag stuff to it. Don't forget and use the card to take photos on though.
If it something is precious, save it in a directory called 'precious' that way you'll know not to accidentally delete it.
Out of all the acorns that are dropped in its life, an Oak tree only needs one of them to take hold and grow. As with the Oak, do the same with your files. Keep copies in lots of places and when you need it again there will be one that has survived somewhere. All that remains is to find it!