Being Linux based anything is possible! Well almost... But the devil is in the detail.

If you want a pretty much 'live' type backup that is running constantly then you'll want something with a low resource footprint and a low latency network connection to some sort of storage medium. TKLBAM is a great backup solution but I don't think it would be ideal for the sort of usage scenario you are talking about. I think a rsync script just monitoring the relevant folders and syncing to a remote storage medium would be the go. You'll also need to consider the database backend. To make this run smoothly, especially if you have a high traffic site, you're going to need some serious grunt I reckon! RAM particularly...

OTOH you could try it with the machine based in a VM with a script to take snapshot after snapshot. And you'll still need some remote storage (otherwise you risk losing your backups if you have a physical HDD failure). You'll also need to consider a script to delete old backups because they'll get used up pretty quick.

Another consideration, may be to utilise the LVM snapshot feature. LVM snapshots are not well documented here on TKL but by my understanding is that it is a realistic theoretical possibility for taking snapshots of a live system. You will need to do some research around that though as it is out of my experience so I can't really help there.

Regardless of what method you use, the reality is that completely eliminating the possibility of some data loss is unrealisitc IMO. I'd go the rsync way myself.