Each website is linked to an application pool in IIS. If the pool is offline, you would get service unavailable messages when trying to browse the site. On the server, open IIS manager, right click the website and open properties. Find out the application pool name. Open the application pools section in IIS manager and find the pool in question.
Under the circumstances you mention, even if you start the pool and try to browse the site, it will die out again. This is mostly due to the fact that the APP Pool user does not have permission to access resources on the server. Basically it is a permissions issue with the pool user. You can try changing the application pool from plesk for the website. Usually this fixes the issue. If this does not work, then you may have to verify the permissions and group memberships of the pool user for the site in question.
Renegade