Yes, you can backup a mySQL database through a
user's cPanel under the mySQL section. Once you load the mySQL section
you will launch the phpMyAdmin for that user. Within the phpMyAdmin
there will be a link called "Export". Click that and select the
database you wish to export and check the box called "Save as File" and
enter in a file name in the text field below. Leaving the default
options checked should be good enough for a standard SQL backup. Then
hit Go.
You will then be prompted by your web browser to save a file to your
local computer. This is your newly created SQL backup file. If
phpMyAdmin fails to produce a file for download and seems to have be
hanging it most likely means that the database you are trying to export
is too big for phpMyAdmin.