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What do you mean by "nothing else works"? Lights? Refrigerator, water pump, etc?
What these things have in common is they use the 12 volt electrical system. The microwave only uses 120 volts so it will work with or without 12 volt power.
Check the voltage on your 12 volt battery. Is it good? If so, do you have a master disconnect switch that shuts off 12 volt power to the house? Sometimes this is called a "salesman switch" because RV salesmen use it when showing a coach to make sure everything is off. Or maybe you have a corroded connection on the battery that's blocking power.
If your battery is run down, it could be your converter isn't operating. It's supposed to supply 12 volts when you're plugged into power or into a generator to supply 12 volts to run stuff and keep the batteries charged.
Let us know what you find out!