According to this tutorial (http://hgusers.com/hostgator-ssh-access/) hostgator shared hosts can get ssh access so 2 of the solutions below assume ssh access ha been setup properly (the setup of ssh is beyond the scope of these forums)
This is also a general solution as most shared hosts nowadays have ssh access
1. using command line
# go to root directory
cd httpdocs
# dump database
mysqldump -u USERNAME -p'PASSWORD' DATABASENAME > database.sql
# zip the whole drupal directory
zip -9 -qr html.zip .
2. Using drush
setup drush (done once)
# get drush
wget http://files.drush.org/drush.phar;
# if above does not work go to and get the link to latest version .phar from
# https://github.com/drush-ops/drush/releases
# wget https://github.com/drush-ops/drush/releases/download/8.1.16/drush.phar
# make it executable
chmod +x drush.phar;
# move it to home directory or any other directory
mv drush.phar ~/drush;
# go to that directory
cd ~
# find out the path to the directory
pwd
# the output would look similar to this
# /home/your_user
# edit the following file
nano .bashrc
# or
# nano .bash_profile
# add this the end of the file
alias drush='/home/your_user/drush';
# use the alias you just setup
source ~/.bashrc;
using drush
# after setting up drush
# go to root directory
cd httpdocs
# dump the database
drush sql-dump > database.sql
# zip the whole drupal directory
zip -9 -qr html.zip .
3. if no ssh present you will need to use cpanel, plesk or another control panel
database backup (one of the following)
- go into cpanel and find phpmyadmin go into your database and start the
export
process
- some panels will have under
databases
a way to backup a database without getting into phpmyadmin
files backup
- go into file manager and find the root directory (public_html or httpdocs) and zip the folder
Notes
More on drupal backup (https://www.drupal.org/docs/7/backing-up-and-migrating-a-site/back-up-your-site-using-the-command-line)
If memory runs out on any of the previous solutions perhaps you ought to upgrade your plan or move to a vps/dedicated/cloud hosting solution
if none of the above works and the support is not willing to help you set a way to backup your site get out of that host they are not worth the money that you spend