Why do 8 electrons form a stable shell? Why not 7? 6? 4? 2?

The modern quantum mechanical electron configuration predicts that electrons will inhabit a shell structure. Beyond the first shell (which can hold 2 electrons) additional shells are complete when they hold 8 electrons in the outermost orbitals (s and p). The "completed outer shell" configuration is very stable, but to get the reason for this stability requires quantum analysis using Schroedinger's equation.

The current model for an atom involves orbitals, namely s, p, d and f orbital. There is room for 2 electrons in the s-orbital and each of the p-orbitals. There are 4 p-orbitals. So they need 8 electrons to be stable. Any less and they "need more". Any more and there is no "room".

Better?

Octet Rule.