Templates are not off limit for api. But it's OFF limit for anonymous user account. In this case, for the default impersonating account ("sitecore\ServicesApi") doesn't have access to the Template items.
Check the image below for the default access level for ServiceApi user account.
In our case, we accessed the api from server side. So, we disabled anonymous access and sent account that has high privilege in the authentication header to read the templates.
And reading template details (section and fields) from Api is an another story. If you start reading it, you will know it.