Assuming one of your questions is "How the back-end architecture of YouTube built to handle this." even though you never used a question mark in your post, the answer to that is that Google is incredibly huge, and has tons of servers dispersed across the globe, and you can be sure that the data is stored on more than one machine, so in case one of them goes down the data can keep on streaming.
Usual backup plans include having off-site backup, but if you need to have high uptime, you may need both off-site and local backups so you can restore quickly from the local, though if there is any disaster ever in the DC you will have to use the off-site ones.
Someone mentioned tape backups, though I'd advise against them, as you don't seem to actually need to have archives of the data, and you probably just want to be able to sync your data to another server to have a backup. There are useful tools like rsync which can keep data in sync and will only upload modified files, sparing you from doing a full backup.
There are ways you can overcomplicate it and ways to burn money setting up a lot of redundancy, but something tells me you can't afford it and you don't need the hassle of managing too many machines.