The Mishna (23a) introduces us to a situation of a karpef she-eino mukaf ledirah – an enclosed area that is not used for living purposes – rather it is a garden or an enclosed courtyard used for storage that is not connected to a house. Although biblically such an area is a reshut ha-yahid – a private domain – the Rabbis of the Talmud ruled that it should be considered a karmelit, where it is forbidden to carry because of a Rabbinic decree.
This is only if the area is larger than a beit se’atayim – the area of land on which two measures (se’ah) of grain can be grown. [Note: A beit se’ah, which can produce one se’ah of grain, is 50 amot by 50 amot. Thus, a beit se’atayim is 500 square amot.] If the area is smaller than this size, then it will be considered a reshut ha-yahid – a private domain – if a number of conditions are met:
According to Rabbi Yehuda ben Bava it needs to have a guard shack or be near the city.
According to Rabbi Yehuda it is enough to have a water cistern of some sort.
According to Rabbi Akiva it does not need any of these things, as long as it is not larger than 70+ amot by 70+amot.
The Gemara explains that the source for the rule of 500 square amot as the maximum for a non-living area is the size of the courtyard of the Mishkan, whose dimensions were 50 x 100 amot (see Shemot 27:18). The amora Rav Yehuda explains that Rabbi Akiva understood the repetition in that passage “the length of the courtyard – a hundred by the cubit, the width – fifty by fifty” as indicating that we are to take the 50 x 50 square and add around it the second 50 x 50 square, giving us a square of just over 70 x 70.
The sides of the resulting square are just under 70 and 2/3 amot. A closer approximation is 70.71 amot (accurate to within one-tenth of a square ama), but even that is only an approximation, as the exact length is an irrational number that cannot be fully calculated. As presented by the Gemara, Rabbi Akiva believes that the approximation of 70 and 2/3 is accurate enough to be accepted by the Sages as the working length, and it is not necessary to attempt a more exact measurement for something that will, in any case, never be precise.