WHM/cPanel backups will copy to a remote FTP. I've never actually used it but it does have the functionality. You would want to watch your disk space though as I don't think it deletes the backups after copying them to the FTP site.
If not WHM/cPanel I'm sure you could find scripts online to do it, maybe even write one yourself.
I'm currently looking at a few applications that may do the trick. I want to use the remote FTP backup to put the backup on my local computer google drive folder. That way I'm not putting it on my actual VPS server.
As far as I know Google drive does not support FTP. Guess it would depend on the FTP server you run on your local machine as to whether or not you could configure it to use the Google drive folder but even if it didn't you could always use something to sync from the FTP folder into your Google drive folder.
Just a word of caution here, most residential ISPs don't like servers being run on their networks.
I have a business account with my ISP, so that will not be an issue. CyberDuck looks like it could be a solution but I haven't had the time to toy around with it. I may just use RackSpace to do my backups.