Both hots show 120 volts to neutral and you will measure 240 hot-to-hot.
Are you saying that only two wires come from the "meter can" into the house? That's a big problem that needs fixing.
Or are you saying that a third wire comes in. Is that third wire at the same potential as your neutral bar? If so, you probably will need to involve the power company. One of the problem may be in the power company wiring. At least one of the problems is in your local wiring. I think you probably want to call your electrician first, but maybe calling the power company first is better. I don't know.
You have an additional problem in that your local ground connection is not good enough.
So probably two problems that need fixing. It may be good that you have two problems because one problem alone might have never been noticed.