There goes the saying, that the Planck length (or similar) might be the shortest physically measurable distance.
Let us wonder, whether this can’t be true. If you take math seriously in nature, already the old Greeks (Pythagoreans) knew, that there can’t be a measure that measures both a side and the diagonal of a square. Which is just the equivalent of the easily proven irrationality of \sqrt{2}.
So, ‘mathematically’ there can’t be such a minimal length.
Also don’t forget, quantum theory doesn’t mean everything being quantized, there are continuous spectra as well.
Anyway, to further solve the puzzle, one might need to investigate possible deviations from the idealized euclidean distance in nature:

• a noncommutativity of the measurements in two directions
• but there must be simpler things also, that just don’t come to my mind right now