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The standard Talmudic measures of length include, for example, Amah, Tefach, and Etzbah. But those measures are pretty loose, around 20% of freedom.
What object or usage, in our Halacha, requires the finest (most precise/exact) measurement, like "2 Amah and 2.549 Etzbah"?
Actually, many of those measurements are precise!
(So, rather than answer your question, I wish to explain how it is based on a wrong assumption.)
In terms of how far one may walk on Shabbos, or how far one may carry something outside an eruv, one uses one's own ammah. Measure from your elbow to the tip of your middle finger. That's your ammah.
So, for example, my oldest son who is 6'2" is permitted to walk further from the city than his 5'3" dad can.
Where these body-based measures become imprecise is when the halakhah has to be applied to a community.
So, taking eruv for an example... Eruv is an interesting case because it is one halakhah that uses the ammah two different ways. A wall in an eruv cannot have a 10 ammah gap, so when it comes to a gaps, bigger measures are a leniency. A lintel (mavui) must run 4 amos to count as a "form of a doorway" (tzuras hapesach), and can't be 20 amos high. So that long measure would also be a stringency in length, but a leniency in height.
In a single eruv, the definition of ammah must be consistent.
More that that, according to the Rashba, the Ritva, and probably Rashi, each law has to be consistent, not just each situation. In other words, all our eruvim must use the same amamah -- even if for one eruv, the larger ammah is a leniency (eg it has gaps) and for another, larger is a stringency (eg the poles are close together).
So, for that eruv lintel, or for sukkah, one consistently uses a narrower estimate. But, since in kelaim (the laws of not planting crops together) most times wide measuring would be the more stringent, one always uses wide measures in kelaim.
Source: Arukh haShulchan Orakh Chaim 363:32-35
Bottom line: the imprecision appears to boil down to demographics -- no two people are alike, but we often need one halachic ruling across an entire community.
