What would happen if there were gravity in Space?

You won't get one here.

Gravity does exist in space. Things are just far enough away from each other than only really massive things (stars, planets) interact visibly.

Gravity is a force exerted by massive bodies.  It exists in space, but only where there is matter.  Empty space can't exert any force, and there is no "up" or "down" in space.  The only reason why we have the concepts of up and down is that we have the Earth's gravity as a point of reference.  Even if space had gravity, it would be the same in all directions, so nothing would fall.

There is in fact gravity in space, and everything in space is in fact falling forever.  A satellite in orbit is in free-fall, falling around the earth but with enough lateral speed to continually miss it.  The earth is free-falling around the sun.  The sun is free-falling around the galaxy.  Meteors striking the atmosphere are in free-fall, but they hit the earth because, as previously implied, they don't have enough lateral velocity to miss it.

Well, there IS gravity in space.... Everything in space IS falling. Everything in space is "falling" towards something more massive. Everything in any orbit is also falling. The definition of being in orbit is falling towards an object but having enough lateral speed to fall past it before striking it, but not so much speed as to escape its gravitational pull. That's not an analogy, it's an actual description of an object being in orbit. Astronauts in space FEEL weightless because they are in constant free fall. Just like a sky diver feels weightless while he's falling towards the ground. They both FEEL weightless, but neither actually is. If there was no gravity in space it would be impossible to go into orbit. Which means the Earth would not be able to orbit the Sun, it would just go flying off into the darkness. The Sun would not be able to orbit the galaxy either. The speed needed to maintain an orbit depends on the mass of what you're orbiting and your distance from it. The space station is orbiting at a speed of about 17,000 mph. But if you were to step out of the space station and fire a retro rocket that would bring your speed down to 0 you would fall straight down from there to the ground, just like if you stepped off of a tall building. This is how spacecraft leave orbit. They fire a retro rocket to slow them down, it slows them down enough that the Earth's gravity (in space) does the rest). It doesn't slow them all the way down to 0 so that they fall straight down, but slow them enough that they just fall out of orbit and re-enter the atmosphere at an angle.

As you've already been told, there is graity in space, for sure.  We can observe its effects over ermarkably large distances.  Sure, planets and other objects are held in orbit by gravitational attraction.  Sure, the sun is held in its orbit around the galactic centre (in the direction of the constellation Sagittarius) by tgravitational attraction.  But there is still more.  Galaxies like the Magellanic Clouds, and the Great Andromeda Galaxy are held in the same cluster as the Milky Way by gravitational attraction.  And beyond galaxy clusters like ours there are super-clusters, also gravitationally bound.

I'm not sure anybody has explained to you why you thought there wasn't gravity in space.  I think  you have in mind experiences such as those astronauts have in space vehicles, where they appear to be floating freely around the vehicle.  The truth of the matter is that both he astronaut and the vehicle are gravitationally bound.  Eery single one, except the Apollo moon vehicles, was gravitationally bound to an orbit around the earth.  The Apollos were either in orbit around the earth, ir orbit around the moon, or in orbit around the sun between one and the other.

So why does gravity seem not to apply to them?  Simply because astronaut and vehicle are in essentially the same orbit.  They are orbiting the earth in parallel, and the difference between the orbit of the astronaut and that of the vehicle is so small we don't normally notice it.  For a big vehicle, though, like the ISS (space station) there can be a difference.  Some of the vehicle will be a little (a few metres) closer to the earth than other parts, and so will 'want' to orbit a little faster.  Because the vehicle is fairly rigid, these various parts all have to orbit together, but it puts some strain on the remotest parts of it.  That effect we call 'microgravity'.  Maybe not the best name, but 'microdifferenceingravity' would be cumbersome.

If there were no gravity, nothing could stay in orbit--in fact there would be no orbits

There IS gravity is space.  That gravity is what make moons orbit around planets, planets orbit aropund suns, and suns orbit around galactic centers.  It is the gravity in space that forms galaxies!

EVERYTHING is space is attracted by gravity to everything else.  It is the BALANCE of all the gravities that makes things work the way they do.

Then planets would orbit stars, and moons would orbit planets.  Oh that is what happens!

Contrary to what you expressed, there "is" gravity in Space, through out. At any point in Space, the gravitational pull of each body near, far and very very far are all there at that point (pulling) in different directions."Fall"as the word you used is the direction of gravity. Things fall in different directions at various points in Space. Since there is no direction in Space that one can identify, which ever direction a body falls is 'down'. There ain't any unversal or common 'bottom' in Space, save the direction in which gravitational pull acts. A body can move indefinitely if nothing obstructs it, against which it hits.

There is gravity in space.  Gravity from any object, even one as small as the earth, extends infinitely in all directions.