Why are there no crystals with 5-fold symmetry?
All unit cells are parallel-sided hexahedra. These are six sided shapes with parallel opposite sides. Their three principle angles may or may not be 90 degrees. And the three side lengths may or may not be equal. All of these unit cells can be uniformly stacked.

Using these building blocks it is only possible to produce planes of reflection, diads (axis of rotational symmetry order 2), triads, tetrads, and hexads.

For example, a cube (all sides the same length, all angles 90 degrees) has diads, triads, and tetrads; plus planes of reflection.
A hexagonal symmetry can be used with 60 / 120 degree angles.

Note that it is impossible to produce a regular arrangement of unit cells to produce a pentad (order 5 symmetry). As @Nathaniel says, this can be almost achieved using Penrose Tiles (2d mathematical constructions), and quasicrystals (real 3d materials). Quasicrystals will produce an x-ray diffraction pattern with a pentad, but the actual atoms do not follow a true 5-fold symmetry.

I suspect your confusion is over the final (macroscopic) crystal shape and the unit cell shape. These are rarely the same. (Salt being an example where they are; but diamonds, garnets, do not)