In principle one can construct a fair die with any number of sides. Let's illustrate with the rather unlikely number 7. Build a shape with square walls and a pentagonal floor and ceiling. Sure enough, it has 7 sides. If the shape is flat, like a coin, heads and tails will win every time. If the shape is tall and skinny, like a pencil with blunt ends, one of the five walls will win every time. By the intermediate value theorem, there is a height, somewhere in between, where all 7 outcomes are equally likely. Number the faces 1 through 7 and you're done. Of course there is no formula for finding that magic height. You probably have to model the tumbling die on a super computer.
If you want 24 possibilities, the above isn't very satisfying. You wind up with a long skinny 22-sided object; not very appealing. Of course you can combine the rolls from a cube and a tetrahedron, 6×4, but there is also a single fair die with 24 faces.
Consider the rhombic cuboctahedron, with its 24 indistinguishable vertices. Take the dual of this object and find a shape with 24 indistinguishable faces. Each face is a strange looking quadrilateral, determined by the centers of a ring of square square square triangle. You can picture this by starting with an octahedron and pushing each face outward, from the center, so that the triangle is partitioned into 3 congruent quadrilaterals that meet in the middle. Push the center of each triangle out just far enough, so that the shape looks round from a distance. The new vertices should be as far from the origin as the original corners.
Scan through the semiregular solids again, and you'll discover fair dice with the following number of sides.
4, 6, 8, 12, 20, 24, 30, 48, 60, 120