Quarks are currently taken to be fundamental particles, so there is no "holds them together"
If by minimum distance, you mean the smallest possible distance in the universe, there are no indication that there is such a thing or any reason for there to be such a thing. It IS likely that we will never be able to measure anything below one Plank length (and currently we can't measure anything below something like 10^20 Plank lengths) but that doesn't mean the Plank length is any kind of minimum distance.
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