Is there a shortest possible distance besides zero? Or there's no limitation at all?
Good question. Check out my discussion of it "http://thisquantumworld.com/topdown.htm" [Broken].
Strictly mathematically, both of your alternatives appear to be incorrect.
There is always a smaller nonzero distance -- just keep dividing by two.
There is also a limit: zero.
But is there a smallest meaningful nonzero distance? That is, some distance that no possible event can occur over a smaller distance, no matter how little time it takes?
Sure. That's called a Planck Length. Planck units are the "smallest meaningful" amounts of anything. Planck time is the smallest meaningful amount of time -- nothing can occur in less time, for example.
The Planck Length is something like 0.0000000000000000000000000000000000016 meters.
That's how far a photon traveling at c will travel in one unit of Planck Time. A proton is about 100,000,000,000,000,000,000 Planck Length units across.
Is there a distance shorter than one Planck Length? Can't you just divide that by two, and so on? Not meaningfully, no. You can conceive of a smaller length philosophically in your head, but that cannot express a valid measurement in the real world.
What will happen when a matter travel for a distance equal to Planck Length? The matter can't travel for any distance between Plack Length and zero, can it?
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