I'm no expert on apertures, but in studying many a camera body, it appears that some apertures are made up of a lot of thinner petals, and can retract into a fairly small area. The actual movement of aperture petals is a rotation inward from the edge, they are designed in such a way that they can fit into small spaces. Think about camera bodies, they aren't very big compared to the lens mount, those petals don't "pull out" towards the edge, they rotate into conveniently compact rings.
I originally thought that the inner "spiking" of the petals wasn't physically real, but inserted to look sinister. However, it appears this is a fairly common aperture design, and quite a few Canon bodies use the technique of spiking the petals after the aperture is stopped up beyond a certain point, though for what purpose I don't know, maybe for a smoother hole and better bokah? Anyway, I just looked at the animation frame-by-frame, and sure enough, it's a continuous model, those spikes aren't coming out from nowhere, they are the actual petals spikes. Of course, it's actually computer generated, but as for being a realistic physical geometry, it works.
No question that those petals could fit fine inside the ring, they're just slightly thinner at their widest point, which is all they would need to be. For an aperture to work, as long as there is a slight overlap of petals at the largest stop, they will work, so you don't have to imagine the petals going much beyond the thickness of their widest point when completely closed. And since the "iris" is behind the rotator ring and chevrons, there's really no concern of interference. their overlap would be such that maybe 4 or so petals would overlap at any given point. If the thickness of each petals is 1 inch thick (they look pretty thin), then you need a 4 inch gap, which is fairly believable.
The biggest problem is machinery and cost. This kind of machined action is requires extreme precision and would be very expensive. Since the liquid is already computer modeled (probably a lot of it is just flat planer effect, ie: After Effects), it makes sense to just throw in a model of an iris and forgo having to make the real thing.
But from what I can see, there is really no physical reason that the iris couldn't be physically built.