Published vs Unpublished
Section 1 - Is my New Site Published?
If visitors to your Site are getting a 404 error like this for your New Google Site ...
( here is a link to a test unpublished Site of mine to try )
... then the most likely explanation is that your Site is not 'published' or that who may view your Site has been restricted (see section 2 below - Who has access to my Site).
To tell if your Site is or is not currently published, got to the New Sites editor for your Site. If your Site is NOT published, you will see a page where the action bar looks like this:
Note that there is no dropdown arrow next to the 'Publish' button.
When your Site is 'published', the action bar at the top of the Sites editor page will look like this:
Note that there is a drop down arrow (the small triangle) next to the publish button which gives you access to additional options for your Site including the option to Unpublish the Site - which will result in visitors to your Site seeing the 404 error message as shown at the top of this article.
If your Site is published and visitors are still getting a 404 error, read the next section
Section 2 - Who has access to my New Site
To see who has access to your published Site, click on the Sharing button:
-------------------------------------
-------------------------------------
A menu will be displayed showing who you have specifically given access to and allowing you to change who has access. There is also a button to change the way links to your Site allow access.
-------------------------------------
-------------------------------------
To allow anyone to find your Site on the web, you need click the Change button to change the Links settings to 'Anyone on the internet can find and open'
-------------------------------------
-------------------------------------
If you do not see an option for 'Anyone can find and view' then your Site is almost certainly in a G Suite domain where the domain administrator has disabled this option and you will need to contact your domain administrator.
Private Sites
If a Site is not shared publicly but is shared with specific Google accounts, users will see the 404 error unless they are signed in with an account that the Site has been shared with. To ensure that users are prompted to sign in to their Google account if they not already, create and use a link in this form:
https://accounts.google.com/AccountChooser?continue=https://SITEADDRESS
(replace SITEADDRESS with the address of the Site)
If not signed the user will be asked to sign in. If signed in the user will be asked to choose an account.