Observation: How'd you like to be the poor NASA employee assigned to answer those calls!
Now, if I was that poor sod, and could get away with it without getting canned, what I'd do is this:
Tomorrow, when the date is correct, either leave the phone off the hook - that would get a few folk worried - or better yet, when you get the caller ringing in, answer, but as they ask their question relating to the end of everything as we know it, have a sound file of a large explosion go off.
Or, just to avoid the nutters anyway, leave a recording going.
Hello, this is NASA. Due to the unexpected end of all human civilisation and the end of the world, if you are hearing this message, it means that within the next 24 hours we will all be dead. Thank you for your call. You won;t be getting charged for ringing us!