What the text says. Luke gives a distance in festival units: "Then returned they unto Jerusalem from the mount called Olivet, which is nigh unto Jerusalem, a sabbath day's journey off" (Acts 1:12, ASV). The underlying command is Exodus: "abide ye every man in his place, let no man go out of his place on the seventh day" (Exodus 16:29, ASV).
The measured limit, and where it came from. The 1915 ISBE sets it out. The phrase "comes from rabbinical usage to indicate the distance a Jew might travel on the Sabbath without transgressing the Law, the command against working on that day being interpreted as including travel". The figure: "The limit set by the rabbis to the Sabbath day's journey was 2,000 cubits from one's house or domicile, which was derived from the statement found in Jos 3:4 that this was the distance between the ark and the people on their march." Roughly a kilometre, or a little over half a mile. ISBE adds the corroboration from Josephus, the Mount of Olives is put at five stadia from Jerusalem in Antiquities and six in Wars, which "would make the distance of the Sabbath day's journey from 1,000 to 1,200 yards, the first agreeing very closely with the 2,000 cubits."
The device for going further, recorded by the tradition itself. The same rabbinic literature that fixes the limit records how it was lawfully extended, and does so without embarrassment: "The rabbis, however, invented a way of increasing this distance without technically infringing the Law, by depositing some food at the 2,000-cubit limit, before the Sabbath, and declaring that spot a temporary domicile. They might then proceed 2,000 cubits from this point without transgressing the Law." The Mishnah's tractate Eruvin is devoted to such arrangements, and its rules are exact to the point of dryness, a traveller overtaken by the Sabbath must say "Let my Sabbath residence be at the trunk of that tree", because "under that tree" is too vague (Eruvin 4:7, as reported by ISBE). ⚠️ This is rabbinic practice as the rabbis themselves record it, not a New Testament rule and not an accusation: the Gospels' Sabbath disputes are about other questions, and nothing here settles them.
Sources: Acts 1:12; Exodus 16:29; Joshua 3:4; Numbers 35:5; Josephus, Antiquities 20.8.6 and Wars 5.2.3; Mishnah Eruvin 4:7; ISBE 1915, "Sabbath Day's Journey".
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