With Focault's pendulum, it is possible to demonstrate in a closed room that the Earth Rotates with respect to an inertial reference frame. And allowing guests to inspect and manipulate the pendulum assembly, I'd say you can convince a skeptic person that the ground beneath their feet is spinning (albeit slowly), if at least this person accepts some version of Occam's Razor.
Is there an experiment, that can be conducted in a closed room that would demonstrate to a skeptic person that the Earth is roughly spherical?
I'd say that Eratosthenes' experiment of measuring light angles neither can be conducted in a closed room nor would prove the earth to be spherical (as the sun could be much closer to a flat earth, or the earth could have a cylinder shape).
Any experiment that would depend on programming a machine (that could be inspected but not reliably audited by a common person) also does not count.
If it requires very fine tuning such as Michelson interferometer, or some hypothetical very high accuracy gradiometer, it could answer the question, but would be nice to give an idea of how the skeptic could be instructed to inspect the device, in order to trust that it hasn't been tampered.
I'm guessing that sadly, there exists no such experiment.
Edit: Note that unlike other questions mentioned, the experiment needs to be conducted in a close room, so no sun observing, ship over the horizon disappearing, airplane watching or traveling allowed!