Carbonite's lowest tier plan, Safe Basic, runs $6.00 /month ($71.99 /year). There are two higher tiers available as well, called Safe Plus and Safe Prime, and run $9.34 /month ($111.99 /year) and $12.50 /month ($149.99 /year), respectively. Each tier is a bit cheaper if two or three years is paid upfront. Plus, they each have a few extras over the base service like external hard drive and mirror image support.
The next page will show a list of all Google products associated with your account whose data can be backed up. All of them will be selected by default, so make sure to unselect all and select only youtube. An easy way would be to untick the "Select All" button at the top which will unselect all items. Then scroll down and check the "YouTube" option.