I think the assumption in your question should be addressed. Franklin Roosevelt and Eleanor Roosevelt were very distant cousins. They were fifth cousins once removed. Their common ancestor was FDR's 4x-great-grandfather, and Eleanor's 5x-great-grandfather, a Dutch immigrant named Claes Martenszen van Rosenvelt who was born in 1623, and died in 1660, more than 200 years before either of them were born.
Or put another way, one of FDR's 64 4x-great-grandparents was one of the 128 5x-great-grandparents of Eleanor. The only reason it's even apparent is by the happenstance that this one shared relative is the one through the direct male line, so they both inherited their surname from him.
In actual fact, FDR was more closely related to Ulysses S. Grant (fourth cousins once removed - one of Grant's great-great-great-grandfathers was one of FDR's 4x-great-grandfathers), and both FDR and Eleanor were more closely related to Martin Van Buren, through different relatives than to each other (both being his third cousin, several times removed).
Note that "removed" just means there is a generation separating the relationship. So your parent's second cousin is your second cousin "once removed". Your grandparent's tenth cousin is your tenth cousin "twice removed".
In that sense, Barack Obama and James Madison may be more closely related by blood than Franklin and Eleanor Roosevelt are to each other. Obama is a shared descendant to one of Madison's great-great-grandmothers, making them third cousins, but "nine times removed" because of the generation gap.
More bluntly, Franklin and Eleanor's blood relationship wasn't very strong. Perfect strangers are often fifth cousins or even closer and won't ever even know it.
One of two things that really allowed them to intermingle was the surname, because they were the direct descendants of the two different sons of that man who died in 1660, those two sons splitting the already considerable family fortune, which each son was able to build upon. As such, 200 years later, these lineages were both still part of New York "high society" so these distant cousins would have some social connections, and there were occasional family reunions.
But the other, probably more important factor that tied them together was an ocean liner voyage. Before either of them were born, Eleanor's grandfather died, and her father Elliott inherited part of his fortune. Shortly after, Elliott took a world cruise. On the Atlantic crossing, Franklin's father James happened to be on the same voyage, and the two Roosevelts (James and Elliott) struck up a friendship which lasted for many years. When FDR was born, James asked Elliott to be the godfather. As friends, mostly, the two elder Roosevelts would have visits back and forth, which allowed Franklin and Eleanor to get acquainted. They were distant "cousins" but it was more out of a shared friendship of their parents than blood that really gave them any connection.
And even then, these visits were relatively infrequent. By 1902, they hadn't seen each other in many years. That summer, Franklin spotted Eleanor on a train, when Eleanor was returning home from finishing school. At that point, both of Eleanor's parents had died, and she and her brother were living with their grandmother, so "home" was Tivoli, New York, which is about 15 miles from where Franklin lived, in Hyde Park.
But as Eleanor recounted, they didn't really re-connect until the year Eleanor "came out" as she put it (in the old-fashioned sense), and was old enough to go out on her own and with friends, which seemed to be 1903. As she explained:
"I never saw him again [after the train encounter] until he began to come to occasional dances the winter I came out and I was asked to a house party at Hyde Park where the other guests were mostly his cousins."
This statement also implies that she never really considered herself to be one of Franklin's "cousins". From that point on, they started corresponding regularly, and they were married in 1905.
For your main question, I am less qualified to answer, but according to one academic article: "in both the US and Europe, the frequency of first-cousin marriage—a practice that had often been favored, especially by elites—sharply declined during the second half of the 19th century". Between 1850 and 1909, 21 of the 48 U.S. states had banned the practice of first-cousins getting married. I am not sure if there have ever been bans on second-cousins; but third-cousins and further away has never had any real law or stigma behind it. But I am no expert on that, so maybe someone else can better fill you in more authoritatively.
But there really wouldn't have been any stigma behind Franklin and Eleanor's marriage, given their relational distance, as fifth cousins once removed. Had they been more closely related, it may have been an issue. Though New York isn't one of the states that had banned first cousin marriage by the time of their wedding, and, in fact, they never have banned it, so maybe they would have been able to get away with it. It's all speculation of course. Being part of New York high society, it seems very unlikely that they would have been willing to suffer the social stigma, which in all likelihood would have been very real, even if New York never actually made it officially against the law.
SOURCES:
A brief biography of their common ancestor, Claes Martenszen van Rosenvelt, can be found in The New Netherland Register, Volume 1, published in 1911.
Another account is found in Roosevelt Genealogy, 1649-1902 by Charles Barney Whittelsey, published in 1902, although that one gets some of the dates and family tree mixed up, which has since been corrected and expanded upon.
The story about the ocean voyage and FDR and Eleanor's subsequent relationship is taken from The Autobiography of Eleanor Roosevelt, first published in 1962. Much of the same information is repeated in the book Eleanor and Franklin by Joseph P. Lash, first published in 1971.
The information about cousin-marriage patterns and law comes from the article "'It's Ok, We're Not Cousins by Blood': The Cousin Marriage Controversy in Historical Perspective" by Diane B. Paul and Hamish G. Spencer, published by the Public Library of Science (PLOS) Biology journal, 2008.