I suggest backing up in two locations. I splurged on a 2TB external hard drive (not just for photos - I'm an architecture major, so if my computer were to crash, years of projects would be gone), and I also have a pro Flickr account where I save all of my photos. I start by just uploading all of my pictures to Flickr, not just my best - then I'll go through and delete that bad ones.
I do recommend backing up online and springing for an account which will allow you to upload high-quality images (many photo-sharing sites have a free account with a storage limit, but for just $6.95 every three months, I have unlimited storage on Flickr... other sites like Photobucket, Smugmug, etc. have similar pro account option) because if you were to damage your external hard drive (spill water on it, drop and break it, it gets damaged in fire, etc.), you'd be able to get your original size images offline. But, I also wouldn't rely solely on an online photo storage site because accessibility could become limited with that, as well (you could lose your Internet connection, the site could have failure, the company could become bankrupt, etc.). Always back up in multiple places, and back up often - you can never back up too often (because when the computer crashes, you almost always lose more work than you think you have).