You can create and host a public website to use with SharePoint Online. This public web site can be accessed anonymously by users.
To create a public web site, select Sites from Office 365 app launcher icon and select Website under Recommended Sites section
The web site will be opened in offline mode which will be available only by the office 365 users. To make it online for anonymous users, click MAKE WEBSITE ONLINE and click Make online button.
You site will be made online after successfully making the site online.
Note: For security reasons, you will not be able to enable anonymous access to other SharePoint online sites.