Some of the more popular online applications for backing up your computer do not yet work well with Linux. Fortunately there are other options that do work with Linux as the companies have taken the time to make sure the Linux users are also included. Since much of the backup utilities are drag and drop with syncing features it's no excuse to not make these applications cross platform.
One of the more popular services is one called Drop Box. This service currently gives away 2 gigabytes of storage space with some page options to get up too 100 gigabytes of space. Even if you do not want to have a paying customer it is 2 free gigabytes to backup your most important documents and pictures.
Ubuntu distributions have the Ubuntu One service that works quite a bit like Drop Box. This works so that Ubuntu users can backup and sync there files on different computers running Ubuntu. The 2 gigabyte version is free with paid options for more. This works well but only with the Ubuntu distribution.
Do not forget to do offline backups as well. Getting USB sticks, SD cards, and backup drives are extremely cheap and should be done on a regular basis. Using online services is great, but if there was a malfunction on their part, it's nice to have another backup.
One of the most important files to backup as a Linux user is your home folder. This is important because it contains your documents as well as your applications settings. Show the hidden folders and back this up on a regular basis in case anything happens. Backing this up will keep all your system settings and your main core files.
Whatever option that you choose, be sure to regularly backup so that you do not lose important files that you might not ever be able to replace.