Half filled and fulfilled atomic orbitals are stable because of : high exchange energy. The problem is with exchange energy. We have learnt that the half and fulfilled orbitals have maximum no. of electrons with the same spin so they can exchange their positions and thus release energy. BUT, we have also learnt that the electrons of the same orbitals have the same energy. Furthermore, the electrons of the same sub shell have the same energy in the absence of magnetic field. so, how on earth do they lose energy?
If I am understanding correctly, your question is actually "how does exchange energy originates?" rather than "is really an atom stable?" so I'll try to answer the former question.
Electrons are not losing energy by exchanging their positions between degenerate spatial orbitals; this is a misconception. Instead, exchange energy is a purely quantum mechanical effect that arises due to two things: (1) the Coulomb term in the atomic/molecular Hamiltonian, and (2) the fact that the electronic wavefunction must be antisymmetric with respect to the exchange of two particles (this is in fact the correct statement for the famous Pauli exclusion principle). The exchange energy is always negative or zero; it depends on a set of integrals which are always negative. However, these integrals are zero for electrons having different spins. Thus, when you have partially filled degenerate shells, having all electrons with the same spin minimizes (makes more negative and thus more stable) the total energy because of the exchange contributions.
My explanation above is very qualitative; if you want a more detailed explanation, just search for "Hartree-Fock method" and if you read about it the concept of exchange energy will become much clearer.