It is physically impossible for the Earth to flip upside-down. Even if it could, you would still not fall into the sky or outer-space. The reason for this is Gravity. In fact, you are falling right now. You are falling towards the center of gravity of the Earth. All mass bodies are attracted to one another. You are a body of mass., and so is the Earth. Since the Earth is much, much more massive, the Earth will always win.
The Earth isn't flat. But your phobia needs addressed.
Something that might comfort you... Think of a rocket ship. Think of how much FORCE it takes for that rocket to leave the Earth's surface. It takes just as much for a ship to leave the North Pole as it does to leave the South Pole, which is technically "upside down" already. So, the Earth cannot flip, but even if it could, if it did you will never be able to exert as much force as that rocket ship, i.e. you cannot leave the surface.
The Earth isn't flat. But your phobia needs addressed.
Something that might comfort you... Think of a rocket ship. Think of how much FORCE it takes for that rocket to leave the Earth's surface. It takes just as much for a ship to leave the North Pole as it does to leave the South Pole, which is technically "upside down" already. So, the Earth cannot flip, but even if it could, if it did you will never be able to exert as much force as that rocket ship, i.e. you cannot leave the surface.
I know it's impossible, but the remote possibility is what scares me most.