Closed Room Experiment to prove Earth isn't flat 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!
 A: If you can convince someone that the earth is rotating and if they believe the earth is flat like a pancake then they probably think that the earth is spinning like a thrown frisbee. But objects tend towards the lowest energy state and that means the disc would evebtually spin along its axis with the highest inertia. So the disc would eventually start spinning along it's distinct axis of rotation like flipping a pancake, which wouldn't be good news for us and it contradicts the weather (clouds would be pushed outwards) and we would also probably see the shadow of one side of the disc cast onto the other in a way that wouldn't be consistent with our night and day cycles.
A: I'm reminded now of one particular consequence of the roundness of the Earth. In the US there are extended plains that during some period of rapid expansion have been divided in rectangular plots. 
The plots are separated by dirt roads, so there is a crossing of roads at every point where the corners of 4 plots meet.
Of course, back then the surveyors could only do local measurements. I assume they would establish the rectangles plot by plot, making sure that each rectangular plot is laid out according to the specification, within the specified margin of error.
These areas with rectangular plots extend over such large distances that the Earth's roundness is significant factor. At some points there is a T-crossing instead of an X-crossing because of the roundness of the Earth.
What I find interesting about this geometry is that all of the individual measurements that went into creating that grid were (presumably) local measurements. (Local in the sense that each plot was too small to notice the roundness of the Earth, but as hundreds of plots are accumulated the surveyors did have to take the Earth's roundness into account.)
None of the underlying measurements was in the form of looking at something at a large distance. So this form of encountering the Earth's roundness is the closest, I think, to meeting the constraint of disallowing measurements that directly span a large distance.
Article by Geoff Manaugh:
Mysterious Detour While Driving? It Could Be Due to the Curvature of the Earth

Photographer Gerco de Ruijter's new project explores the places where
  our highway system goes astray, thanks to the challenges of imposing a
  rectilinear grid onto the spherical surface of the planet.

LATER EDIT  
After some more reading:
The planning of the Jefferson Grid was to survey plots of land in such a way that the east and west boundaries would be parallel to the longitude lines. The aim was to give all plots of land equal area, not necessarily the exact same length and width. As you move North the longitude lines converge so in order to maintain equal areas of the plots the rectangles are made longer. Every 24 miles there is a correction where the number of plots per degree of longitude is adjusted, thus keeping the width above a specified minimum. I imagine that to the south of such a correction zone 100 plots of land align with 99 plots of land to the north of it. Therefore in those every-24-miles correction zones the north-south roads generally don't line up. 
