Sql Server Management Studio 2008 has suddenly become unresponsive

This might be a very simplistic answer, but this happened to me because a dialog it was attempting to present was off of the desktop and/or hidden behind something else.

When it beeps like that, can you do the whole "Alt-Space m" and move something into focus?

The 'Connect to Server' window is off-screen. If you use Alt+F4, it will close the dialogue but keep Management Studio open. After this you can initiate a new dialogue through the menu and connect to the desired server.

After closing, the 'Connect to Server' window should appear inside of the main window the next time you start the program.