Q:

Why were the Night's Watch required to be celibate?

If the Night's Watch was supposed to last thousands of years, and was given land, why not have them raise families in the Gift and send all adult males to guard the wall on a rotation?

A:

Because love is the death of duty.
If they have families, they will lose the focus on the defense of the human realms against the Others. And when the time comes when they must choose, they will always choose their families rather than the realm. That is why they give up everything, to be completely focused on one duty.
Maester Aemon explained that part of the vow in detail in his conversation with Jon when it was suspected that Jon would desert for love of his family:

“Jon, did you ever wonder why the men of the Night’s Watch take no
wives and father no children?” Maester Aemon asked.
Jon shrugged. “No.” He scattered more meat. The fingers of his left
hand were slimy with blood, and his right throbbed from the weight of
the bucket.
“So they will not love,” the old man answered, “for love is the bane
of honor, the death of duty.”
That did not sound right to Jon, yet he said nothing. The maester was
a hundred years old, and a high officer of the Night’s Watch; it was
not his place to contradict him.
The old man seemed to sense his doubts. “Tell me, Jon, if the day
should ever come when your lord father must needs choose between honor
on the one hand and those he loves on the other, what would he do?"
Jon hesitated. He wanted to say that Lord Eddard would never dishonor
himself, not even for love, yet inside a small sly voice whispered, He
fathered a bastard, where was the honor in that? And your mother, what
of his duty to her, he will not even say her name.
“He would do whatever was right,” he said... ringingly, to make up for
his hesitation. “No matter what.”
“Then Lord Eddard is a man in ten thousand. Most of us are not so
strong. What is honor compared to a woman’s love? What is duty against
the feel of a newborn son in your arms... or the memory of a brother’s
smile? Wind and words. Wind and words. We are only human, and the gods
have fashioned us for love. That is our great glory, and our great
tragedy.
The men who formed the Night’s Watch knew that only their courage
shielded the realm from the darkness to the north. They knew they must
have no divided loyalties to weaken their resolve. So they vowed they
would have no wives nor children. Yet brothers they had, and sisters.
Mothers who gave them birth, fathers who gave them names. They came
from a hundred quarrelsome kingdoms, and they knew times may change,
but men do not. So they pledged as well that the Night’s Watch would
take no part in the battles of the realms it guarded.
“They kept their pledge. When Aegon slew Black Harren and claimed his
kingdom, Harren’s brother was Lord Commander on the Wall, with ten
thousand swords to hand. He did not march. In the days when the Seven
Kingdoms were seven kingdoms, not a generation passed that three or
four of them were not at war. The Watch took no part. When the Andals
crossed the narrow sea and swept away the kingdoms of the First Men,
the sons of the fallen kings held true to their vows and remained at
their posts. So it has always been, for years beyond counting. Such is
the price of honor.
A craven can be as brave as any man, when there is nothing to fear.
And we all do our duty, when there is no cost to it. How easy it seems
then, to walk the path of honor. Yet soon or late in every man’s life
comes a day when it is not easy, a day when he must choose.”AGOT - Jon VIII

This is a great passage which really explains the two most troubling vows of the Black Brothers. First, the vow of celibacy. Second, the vow of neutrality. They must remain celibate for if they have wives and children, their loyalties shall be divided between safety of their loved ones or safety of the realm. And then neutrality because once again, the Black Brothers must not feel any divided loyalties between defense of their former homes and families and defense of the realm.
Until the day of choosing comes, every man is brave and honourable, every man keeps his vows. But that one day comes, as surely as the day he was born, when he must choose. Would he choose his sons? The love of his life? Or the people of the realm? Not every man is strong enough to choose the latter. So they must eliminate the complications to make the vows relatively easier for the Black brothers.
The Gift and the new Gift have in fact been inhabited by normal folks, normal families. And they are occasionally called to defense of the Wall when need be, like Jon called Mole's Town people to Castle Black when Thenns attacked. But they are not Black Brothers. When fight seems hopeless, their first priority will be their families, and they will flee. Just like the Mole Town folks did before Battle of the Wall. Even if we ignore that, if Black Brothers were allowed to own property, start families, the Watch would become eventually a Kingdom in itself, rather than the shield and defenders of all the Kingdoms. And that will sooner or later bring them in political conflict with their neighbouring Kingdoms. Even as things stand, many mad Lord commanders tried to make the watch hereditary, imagine what would happen if there were Houses, families and clans on the Wall as well. They must stand united, all the men of the seven Kingdoms, in defense of all the Kingdoms while staying out of their internal politics.
Jon experienced the truth of Maester Aemon's words despite harbouring doubts initially. When he fell in love with Ygritte, his duty started becoming harder and harder for him. But when the day of choosing came for him at Queenscrown, Jon chose duty and abandoned Ygritte, as much as it hurt him or haunts him still. And yet he failed in some ways again, when he spotted Ygritte while defending Castle Black, his hands won't release the arrow and thus he let Ygritte slip by, simply because he loved her. Who knows how many brothers of Night's Watch Ygritte killed before someone else shot her down?

A:

They weren't. They were required to "take no wives and father no children". Even before contraceptives, there are ways of having sex which have no possibility of pregnancy, and that doesn't even count homosexual encounters.
While celibacy is a way to ensure that, it isn't necessary. It was shown in the novels and books that the members routinely visited brothels, and that this was tolerated, if only unofficially. Maester Aemon made the comment that

If we beheaded every ranger who lay with a girl, the Wall would be manned by headless men.

From a practical standpoint, despite Aemon's poetic explanation, there's also a more practical one; children mean heirs, family, interconnections with other families in the Watch and with people from the South. And at some point, humans being humans, you'll have the equivalent of one or more Houses arising among the Night Watch, and then once that happens, what's the impetus for them to remain neutral in the politics in the South? Having connections means there's a reason to interfering with what's happening down South. If you're Lord Commander of the Watch and your daughter is married to a Mormont and the Mormonts are getting grief from the Boltons, wouldn't you be tempted to flex some muscle to sort that lot out, even just as a threat?
Without children, there's no inheritance. There's none of those interconnections.

A:

Because it generates a conflict of interest that the Watch would rather not have present.

Maester Aemon: Did you ever wonder why the men of the Night's Watch take no wives and father no children? So they will not love. Love is the death of duty."
Game of Thrones - S01E09: Baelor

You may think that having something to defend would make them better at their jobs, but think about this: What about the sons born who'd rather go south? Should they be bound to the wall for the sins of their fathers? If not, then suddenly you have a bunch of Night's Watch who are faced with the same decision that caused Jon to run away for a night. This problem is mitigated by just not letting them form families. 

