Could a non-pointlike structure of elementary particles explain their spin? Elementary particles are considered point-like. Which makes the concept of spin rather problematic. It can neatly be described by group theoretical considerations, but that offers no explanation what it actually is.
But what if we consider particles to be extended particles, their extension reaching into small extra spatial dimensions? For example, in the one-dimensional case, that the particle is a small circle structure around a thin Planck-sized cylinder, which would have the nice property of a Lorenz invariant Planck length, as the extension is perpendicular to the 1d bulk space. This could be extended to 3d if we consider three small curled up dimension (each to a Planck length circle).
In 1d, the circle could rotate in two opposite directions, and spin would correspond to actual rotation.
 A: Here's an intuitive way to think about spin. Consider a spinning ball. It has an axis with a north and south pole. Now there are two invariant tangent planes on this spinning ball. The ones on the north and south poles. All other tangent planes move.
Now let us draw an arrow on the tangent plane on the north pole. Obviously this will spin along with the spinning ball.
Now let us shrink the ball. The two invariant planes will remain invariant and as we shrink to a point we can imagine both are still there. But they are infinitely close to each other.
So we need to think how the spinning arrow should accomodate this. After all, which plane should it be on? Well, the obvious thing is to say both! And we arrange this by the arrow first spinning on one plane and then on the other.
This gives us a double cover of the rotation group and so we have recovered spin.
A: I agree that intrinsic spin is somewhat unintuitive, but it really is point-like.
Consider a disc spinning with constant angular speed in its plane, with the axis at its centre. This disc isn't made of atoms, it's just an idealised geometrical object, a set of ideal zero-dimensional points. All of the points in the disc except for the centre point have orbital angular motion, but the centre point just has pure intrinsic spin, not orbital.
In our mathematical model of particles, the intrinsic spin of a fundamental particle behaves exactly like that centre point. It's not easy to visualise, orbital angular motion keeps trying to sneak into our visualisation. So don't trust the visualisation, trust the mathematics. ;)
