Why can the SA account no longer access the master database?
We know you are getting 18456 errors. The message (other than the 'Login failed' part) and the error state are much more important for diagnosing. In the SQL Server error log, there will be more information than you get from the application or the dialog in SSMS. In SSMS open Object Explorer for the server in question, expand Management, expand SQL Server Logs, right-click "Current - ..." and choose View SQL Server Log. You should be able to find events like this:



This number and the extended reason are not reflected elsewhere because it's meant to obscure the actual failure from the end user, in case that user is malicious. (For example, the state for wrong password might help them see that they are on the right track.) 

You can see a list of all the states I know here, which should help you resolve the issue: 


https://sqlblog.org/2011/01/14/troubleshooting-error-18456


For state 16 it doesn't make sense that the database is offline or inaccessible, unless someone has somehow demoted sa from the sysadmin role and/or explicitly denied access to master, since master can't be offline. I suspect it is much more likely that their default database has changed, or that the database they tried to connect to explicitly is not online. What happens if you run:

ALTER LOGIN sa WITH DEFAULT_DATABASE = master;


Also, why is your application using the sa login? Have you considered creating a separate, less-privileged account to dedicate to your application?