What would happen?

I'm with you, man

Well, falling is "go down" because of the gravity.
The thing is, in outter space, there is no "up" or "down". So objects do not " fall". They certantly move, but not fall :p
(Physics exams are taking over my mind)

I can count to ten.

Haha I love all of you :P

@darkan
"Down" is a relative term in relation to gravity (or whatever direction the acceleration is strongest). Space has gravity. Therefore space can have a "down" direction. Si?

"Down" is a relative term to whatever your reference point and direction in space and time is.
For example, if you are in a plane and that plane "is going down", if you take the earth as reference, then you are going down, too. But if you take the plane as reference, you are not moving at all.
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In space, it's said that there is no up or down, because there is not a clear or common point of reference. Here on earth, pretty much always and everyone takes the earth as reference. But in space, it's not that simple.

In our solar system we could think of the sun as "down", because it is the point of greatest gravity in the solar system.

You could, but it would end up being impractical in most situations

I still maintain the point that in absence of any other point of reference, "down" should be in the direction of the net acceleration vector.

Oh, this reminds me of Ender's Game (book) where Ender tells everyone the enemy gate is "down".

Yes. And they all turn themselfs to place the gate under their feet, making themselfs the point of reference.
And you can't say "down" without point of reference.