It is shown in end that humans are living on a spaceship orbiting near saturn. But when was this ship built? There was no mention of spaceship on such a large scale being mentioned.
In The Science of Interstellar by physicist Kip Thorne, chapter 31 talks about them using their new theory of quantum gravity, discovered by Murph with help from the "quantum data" transmitted by Cooper (see my answer here for details on the physics ideas involved), to temporarily decrease the Earth's gravity and launch the huge colonies off the Earth:
Early in Interstellar, when Cooper first visits the NASA facility, he is shown a giant, cylindrical enclosure being constructed to carry thousands of humans into space and house them for many generations: a space colony. And he's told there are others being constructed elsewhere.
"How does it get off Earth?" Cooper asks the Professor. "Those first gravitational anomalies changed everything," the Professor replies. "Suddenly we knew that harnessing gravity was real. So I started working on the theory—and we started building this station."
How did it get lifted into space? The key, of course, was the quantum data (in my scientist's interpretation, the quantum gravity laws) that TARS extracted from Gargantua's singularity (Chapters 26 and 28) and Cooper transmitted to Murph (Chapter 30).
Murph must have figured out how to reduce Newton's gravitational constant G inside the Earth ... In my interpretation, with Newton's G reduced inside the Earth to, say, a thousandth of its normal value for, say, an hour, rocket engines could lift the enormous colonies into space.
As a byproduct, in my interpretation the Earth's core—no longer compressed by the enormous weight of the planet above—must have sprung outward, pushing the Earth's surface upward. Gigantic earthquakes and tsunamis must have followed, wreaking havoc on Earth as the colonies soared into space, a terrible price for the Earth to pay on top of its blight-driven catastrophe. When Newton's G was restored to its normal strength, the Earth must have shrunk back to its normal size, wreaking more earthquake and tsunami havoc.
But humanity was saved. And Cooper and ninety-four-year-old Murph were reunited. Then Cooper set out in search of Amelia Brand in the far reaches of the universe.