I am getting exactly the same, occasionally, on a test server I have set up (Windows Serevr 2008 R2, FMSA12). Once it happens, the only solution is to restart FileMaker Server.
I wish it was that easy.. Like I said... I have even completely removed IIS and FileMaker and started it from scratch with no luck. Like I said the database is visible in the web page but data will not be visible and I get the "Database is not available error".