Eruvin, Chapter 4, Mishnah 3. This Mishnah deals with someone who went outside his Shabbos boundary with permission - meaning, in a manner that halachah allowed.
What are the cases in which a person goes out with permission?
One who goes out to testify that he saw the new moon, for the purpose of establishing Rosh Chodesh - something a person is permitted to go out for.
One who goes out to save people from a fire.
One who goes out to save people from a marauding troop - cases in which the Sages permitted going out because of the danger, or in which halachah established that it is permitted.
"Ve'amru lo kevar na'asah ma'aseh" - and they told him that the matter had already been taken care of:
After he went out, he is informed that the matter has already been handled and there is no longer any need for him: the troop has moved on, the fire has been extinguished, others have already testified about Rosh Chodesh and it has been sanctified. Even though someone who goes outside his boundary ordinarily has only four cubits, here, since he went out with permission, the Sages said that he has "alpayim amah lechol ruach" - two thousand cubits in every direction - from the place where he learned that he is no longer needed.
"Im hayah besoch hatechum - ke'ilu lo yatza" - if he was within the boundary, it is as though he had not gone out:
And if, despite having left his original boundary, the new two thousand cubits measured from where he is standing overlap with the two thousand cubits from which he set out, we view him as though he had not gone out at all. He may return to his original town and be like one of the residents of that town, who has two thousand cubits in every direction beyond the town.
"Kol hayotz'in lehatzil chozrin limkoman" - all who go out to save lives may return to their place:
The Mishnah explains that the basis of this leniency is derived from another leniency that we find: anyone who goes outside his boundary in order to save others is permitted to return to his place. And in detail: even if they went out with their weapons, they are permitted to return and bring them along. So we see that when it comes to those who go out to save lives, the Sages were exceedingly lenient, to the point of permitting the return trip with the weapons.
In summary: On the strength of this leniency, they permitted him two thousand cubits from the place where he learned that he was no longer needed; and as long as those two thousand cubits overlap with his original two thousand cubits, he may return into the town and be there like one of its residents, receiving two thousand cubits from the edge of the town.