There could be the multiple reasons behind this.
Please check the following steps to find out your issue and solve it.
Step1: Remove the maintenance.flag file from the root folder of your Magento.
Step 2: Rename/Remove the .htaccess file temporary which is placed on the root folder of your Magento and then check if 503 error is gone or not. If 503 error is not coming after renaming/removing the file, that means there is something missing in the server configuration. (e.g. Wrong PHP version).
Step 3: If 503 error is still appearing after renaming/removing the .htaccess file, then it could be the issue of file permission.
Run the following command from the root folder your Magento to give 644 permission to your index.php file.
chmod 644 -R index.php
if it doesn't work then try to run the command with sudo
sudo chmod 644 -R index.php
Note: Don't forget to put the .htaccess file again to the root folder.
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