I'm trying to use my google script app in the google site which should be publicly available to anyone. When I'm trying to deploy my google script as a web app (Publish->Deploy as web app), in the "Who has access to the app" field, the "Anyone, Even Anonymous" option is not available. I have only "Only myself" and "Anyone within my domain".
In order to let users in the domain to share the webapp with "Anyone, even anonymous", your domain Admin should do the following:
Beware, because of a current bug, if you created the script when the sharing settings were OFF, you won't be able to share it outside the domain, even if you change that. The sharing settings are applied when the script is created, and those settings cannot be changed. A new script should be created for that (after enabling sharing outside the domain).