It's complicated.
That rotating disc would in fact produce artificial gravity, directed outward from the center of the disc, so the arrangement they had with people climbing down ladders to rooms set along the outer wall of the disc would work.
However, there's a problem with Coriolis force—basically the effects of your feet moving faster than your head as you stand on the floor of one of those rooms. Your inner ear perceives that, and the usual effect is strong nausea. The only way to combat this is to have a disc or ring that rotates slowly—about two times a minute is the threshold I've heard. To get one full Earth gravity, that means the ring has to be 250 meters across. That's 820 feet, so to get from the center to the "floor" you'd have to descend 410 feet through gradually increasing gravity—like taking the stairs from the top of a 40-story building, which means there would have to be elevators.
Of course, we don't know how much artificial gravity would be just enough to prevent the damage caused by long-term weightlessness (bone loss, muscle atrophy, other more subtle effects). Maybe one-third gravity (like that on Mars) would be enough. That would reduce the diameter of ring that was needed, but it would still be quite large—something on the order of 80 meters or 262 feet. (I found a handy graph and the actual equations here:
https://en.wikipedia.org/wiki/Artificial_gravity.) Obviously the ring in the spacecraft in The Martian was nothing like that large. So they fudged it. The principle is right, but the engineering to make it practical would be enormously complicated.
That's why there isn't one of those rings on the ISS.
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