You’ve understood correctly that all the planets are level with each other. They all move in circles (orbits) around the sun. There are two sides to that circle, one of which you could say is “above” and one is “below”.
So how to decide which way is “up”? We use the direction of rotation. When you look from above, the planets should rotate anticlockwise. (This is also known as the right hand rule.) Using the North/South poles of the sun’s magnetic field wouldn’t work so well, because they flip every 11 years.
If you look “up” or “down” you won’t see any objects from our solar system, but you can see many things that are outside it. Stars within our galaxy (the milky way) are mostly aligned in a stripe across the sky. Galaxies that are further away contain many more stars, but each galaxy appears as just a single point of light. As far as I understand, galaxies are randomly distributed in all directions.
Not only is there no universal up and down in the universe, there is no forward or backward, and no left or right. There is also no universal reference to measure your speed.
All of these measurements are relative. They have no meaning until you designate a reference from which you are measuring. Once you declare that “that way” is up, then the other way becomes down. If you declare that the sun is stationary, then the speed of everything is determined by measuring how fast things are moving towards/away from the sun. But if you changed the reference to be the center of the milky way galaxy, then the speed of everything would suddenly be different.
Even in your question, we customarily think of the north pole as “up” and the south pole as “down”, mostly because we’re used to seeing maps and globes depicted that way. But there’s no objective up and down in the universe.
Regarding your specific question about our solar system, all of the planets formed in the same general plane, so if you were on the north or south pole and you looked radially outward from that point, you would not see any planets. All of the planets are basically lined up and moving in the same plane, because that’s how solar systems generally form.
I suggest you watch a video on youtube of how the sun and the planets really move through the galaxy. Once you do, you should be able to deduce that what’s “below” Antarctica is just what’s “ahead” or “behind” our solar system in our journey through the galaxy, which yes, is a bunch of stars and other stuff.
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