Publishing

Step 1: When you feel your Google Site is ready to be published for your viewing audience > click "Publish" in the blue box 

Note: Google automatically saves your changes, like a Google Doc, but for others to view your changes/updates, click "Publish"

Unpublishing

Step 1: If for whatever reason you need to take down your Google Site, for overall curation or mass revisions,

Step 2: Click on the drop-down arrow next to "Publish" 

Step 3: Then click on "Unpublish" 

Once done editing, publish your site again for viewing