Once SDL contract is over, what happens to the current running site in
production? Will that stop working?
Yes (if the question tags are correct). But, even if it continues running, having it that way would be using unlicensed software which has legal implications. Contact SDL (RWS) directly to see what your options are.
See also Nuno's comment below. It's based on your license type.
Wondering just taking the backup of the database is sufficient or
there is any script I need to run to get all the content/binaries from
the databases. Please suggest.
Sufficient for what? All content including binaries, if managed through Tridion, are contained within your CM DB. Having a backup of said DB ensures that you have a snapshot of the CMS's data structure containing all the content.
You can have binaries not in the CM database (but in for example Azure), but that was not an option in 2013 IIRC.