When I bought my house, an old 300 gallon tank was in my basement full of #2 oil or heating fuel. I now have natural gas furnace. What can I do with the oil? The tank? Can I use it in my kerosene heater? Sell it? How long will the fuel last in a 300 gallon tank 3/4 full?

No do not use it in your kerosene heater unless you want to really stink up your home and have a professional into clean and repaint the interior. #2 is a very dirty fuel as compared to kerosene.

The fuel will be in the tank just about forever until you remove it. Call a local fuel oil dealer to see if they offer a pump out service.