So my system is twofold. I save them on an external and on CD's.
Is this overkill? Well, unfortunately, I learned the hard way that hard drives sometimes crash. I backed up many of the important pictures on CD's before that happened... but not all of them. I'm honestly not sure if the pictures will be able to be retrieved. It depends on the condition of my hard drive.
If they're photos you can stand to lose, I guess you don't have to back them up. But if they're important enough to save in one place, my guess is they're important enough to save in two just to be safe.
Flickr, I honestly don't know if that's safe or wise. CD's seem like a safer option to me. Or a second hard drive. Or both. Back up your back up to your back up.