You can use a domain name like example.com, that you already own for your Google site.
There are Two Types of Google Sites,
-> Classic Site
-> New google Site
Adding a domain name in Classic site is much more simple and Easy, than compared to google sites, and it has an inbuilt feature that allows us to add a custom domain through a web address option ,you can see the full instruction on how to add a domain name to google sites by clicking here.
Then the next one is new google sites, although the designing of the website is much simpler in this new google sites, it comes with a major disadvantage that , it does have an inbuilt facility like the old google site, to add a custom domain name to this google site.
so, to add a domain to this site is not possible , but we can forward or redirect the domain to point to our new google site.
For that go to your host account, iam using godaddy here,
-> In that click on the manage domain opiton,
-> Then, Click a domain name in the list .
-> Then Scroll down and Select MANAGE DNS .
– > Then edit the CNAME record to ghs.google.com
– > Then , On the DNS Management page, scroll down to the Forwarding section.
-> Next to Domain or Subdomain, click Add, and enter your domain or subdomain name:
-> Then Fill in the necessary Details,
– > Under Forward to, select http:// or https://
– > Enter the URL you want to forward your domain or subdomain to.
– > Select your Forward type : Permanent (301) or Temporary (302).
-> Select your Settings : Forward only (i recommend this for google sites)
– > Click Save
NOTE : It may take 24-48 hours for your domain to forward to an existing site.