According to Pauli’s exclusion principle, an $s$ orbital contains at most two electrons with the opposite spin (up and down). Why can't an $s$ orbital contain a third electron whose state is the linear combination of spin up and down?
Because we can have only one electron per quantum state. Spin up and spin down are two different states. A linear combination of the two is not a new independent state. It is obviously formed from the spin up and spin down states.