I don't think cPanel allows access to the backups other than to be able to restore them.
 
For mysql, you could setup a cron to create a backup that you could download.
 
  Code:
  
mysqldump --opt -u <cPanel user> -p <cPanel pass> <database name> >backup.sql
  So if your cPanel login was
andychev
$3cr37
 
and your database was 
main (which in reality is called andychev_main)
 
You would do this:
  Code:
  
mysqldump  --opt  -u andychev -p $3cr37 andychev_main>/home/andychev/www/backup.sql
  You might want to consider creating a mysql user with read access to the database and use that instead of having your cpanel info sitting in a cron file, though its probably not a real big deal.