If you are going to go paperless in your life, you best be backed up when you do. Since there's no paper backup filed away in case the hard drive fails, the building burns or the flood happens, 100% reliable backup becomes crucial.
Years ago, in my quest to go paperless at home and in the office, I researched all kinds of alternatives from USB hard drives to tape to DVDs. Each had their problems with failures, difficulty of use, and practicality. For a while, I used two Maxtor 300 Gb USB hard drives. One would be online and backing up every night, the other stowed away safely in the fireproof computer media rated safe. Much to my amazement and disappointment, the drives continuously failed. After several returns and replacements from Maxtor, I determined a) that these drives just aren't meant to be run for a week at a time every other week and b) Maxtor makes crappy drives, and finally c) perhaps this is the reason Maxtor is no more...
I also tried burning DVDs each week and storing them in the safe. This proved to be difficult to keep up with and time consuming. Backups in a paperless world are too important to do only weekly and they must be done rigorously.
For several years, I used a service for online backups called Mozy. It worked fairly well creating backups, until I actually tried to restore some files after a hard drive failure. Then it didn't do very well at all. After hours of working with Mozy staff to get my files back, I started looking for other solutions.
When Dropbox came out as a product, I switched over it it, and have found it to be very reliable and easy to use. It gives you access to your files on all your devices and keeps a version of your files ON the device if you want, so you can work without a connection. Over time, I moved all my files, music and pictures to Dropbox.
I am also using and testing Microsoft OneDrive Pro and waiting for it to have features similar to Dropbox. Right now, it is does not have all the features of Dropbox, and it doesn't sync consistently. So I'll stick with Dropbox until it has the features I need and is more reliable.
I do NOT recommend using free storage such as Microsoft's free OneDrive or Google's free Docs. I don't recommend any free service for that matter. After reading the Terms of Service (TOS), you'll find that your files aren't safe or even yours.