as they were complaining about an oscillation, at a guess, I'd say a few steps, or about a quarter of a time around. Skylab was about 1000 times as massive as an astronaut, and there are about 2000 10 arc minutes in a circle

@JCRM thanks, I didn't ask about reproducing the problem, only about simple rotation. But that's an interesting question, feel free to ask it separately!

NASA didn't see a rotation, they saw an oscillation. So an answer would have to be done from first principles, which means working out how much force a runner exerts in a tangential direction by pushing off on each step, and how much of that is cancelled out at the end of each step. This in turn involves aerodynamics of the human body. More biology than space exploration...

@RoryAlsop I don't believe that's right, at least not to get a reasonably good answer. If a 65 kg mass rotates once around a given axis at a distance of say 2 meters from it, the station will rotate a tiny bit in the other direction. It's a simple ratio of moments of inertia. It doesn't matter how fast or slow, time doesn't enter in to the problem. If you can find Skylab's moment of inertia around its long axis, you've 90% solved the problem. All you need is the radius of the jogging track and a typical human height, mass, and center of gravity estimate.

@Hobbes See comment above. Note that I have asked about two astronauts and we can assume they are running opposite each other.

Folks: the question is clear, unambiguous, and answerable. "Question: How many times would two astronauts have to run around on this track to turn Skylab by 10 arc minutes around its long axis?" Unfortunately some comments are trying to take the question in a different direction, suggesting I mean something different than what I've written. Remember that a rotation around the long axis will throw the other two axes out of alignment, which is why I've included tSkylab's tolerance of 10 arc minutes.

so, basically a simple maths question, not really related to space exploration - or are you asking "what was Skylab's moment of inetria?"?

@JCRM No. If someone doesn't want to handle the complexities of the torque that the running induces with respect to the possible 90 minute rotation of the spacecraft as it keeps one axis pointed to the nadir, I've allowed for that kind of answer in the question. See the two sentences following the sentence labeled Question: However a full answer based on the possible attitude rotation of the spacecraft relative to the orbit is also more than welcome. So no. Upon reading it, it's evident that it's not "basically a simple maths question". Skylab operated somewhat differently than the ISS.

@JCRM the linked NASA Technical Memorandum explains more about the various ways that Skylab could operate; the different attitudes and rotation states.

you're explicitly asking about rotation NOT oscillation, which means the attitude control would have to be disabled.

@JCRM yes indeed; it's like the joke "Q: How many psychologists does it take to change a light bulb? A: Only one, but the the light bulb has to want to change". In this case we can assume the two astronauts are running in order to perform the rotation intentionally.

about a quarter turn. in about a second. The orbital period doesn't really come into it.

@JCRM It is necessary (but not necessarily sufficient) to have a moment of inertia about the long axis to do the calculation. What value did you use for that? I think it will take a lot longer than that, and so the orbital period may come into play, depending on Skylabs attitude and rotation state.

You're right. It would take longer tahn that - I misremembered moments of inertia (good job this isn't a Maths stack exchange) It's about five eighths of a circuit, so about 5 seconds.

Let's please answer the question as given in the title and Question: line. I find that question to be clear and reasonable. The quotation from the technical memorandum was presumably given to assist those writing an answer. If you think it contradicts with the premise of the question, then either ignore it when writing your answer, or write your own question.