Anonymous access is only accepted in the public web site in SharePoint Online. In the private site, such as Team Site, there is no Anonymous Access button in the Permission Tools. It is not possible to have a regular private site collection in SharePoint Online be available for anonymous access. Please note there is only ONE public site for each tenant like “company-public.sharepoint.com” that you can use for public access.
The procedure to enable the anonymous access to this public site is just publish this public site by clicking the top right link “ ”.
When the public website is online, anyone from internet can view the entire site. You can verify this from permission setting as screenshot below.
You could disable anonymous access to this public site any time by making the website offline. When the public website is offline, it cannot be viewed or accessed publicly on the Internet. Instead, visitors to the site will see a login screen. Only authenticated users, like the website owner, invited designers or members can view and customize the site. This is the default experience in Office 365, which is usually how you want it while designing and customizing a new site. The reason is because you don’t want a half-designed website containing placeholder or missing content to be viewed by the public, including your customers. It’s a better idea to first complete the website then make it online.
In additional, you could make some pages not available to anonymous access by unpublishing them. Publishing and unpublishing pages is a convenient and more granular way to control when an individual page can be seen by everyone.
Regardless of the SharePoint Online public web site online or offline, the permission granted to authenticated users will be able to access the site based on their permission. The online or offline configuration only enable or disable anonymous access.
I'm still in the process to evaluate the compliance center report and eDiscovery search against the anonymous access users. If you have any experience on this, please let me know.