I'm a lead developer on Genie Timeline, and I'm excited to say that we're really, really close to our own cloud-based backup solution! We've been working with several big names in cloud storage, and should have a version of Timeline with online support by the end of the year.
If you like, I'll post an update as soon as we have a beta or initial release available.
We appreciate that no matter how easy we make Genie Timeline to use, so long as people need to first buy an external drive then remember to plug it in each and every time they want to back up their system, backup will remain a real pain in the you-know-where. We hope that a cloud-based Genie Timeline solution will make being too busy to back up a thing of the past.
Now, in response to your initial query, Timeline will theoretically work if you use a cloud-backed storage instead of an external USB disk or network share. Performance will be sub-optimal, seeing as Timeline won't actually know it's writing to the internet instead of a local disk, but you can use a method such as the one described here to get your SkyDrive to be seen by Windows as a mapped network drive; after which a Timeline backup is only a click away!