Suppose we have two hydrogen atoms in the ground state with spin of both electrons pointing upwards. Then the two electrons are in the same state. This should be against the exclusion principle. Now suppose we have 1 mole of hydrogen atoms in a chamber. Certainly, most of them will be in the ground state (at sufficiently low temperature), and among any three of those in the ground state, at least two will have spin in the same directions, hence the two electrons are in the same state. How is the exclusion principle valid for those two electrons?
My doubt is mostly about which parameters determine a "state". Suppose two different hydrogen atoms having the same quantum numbers are in different points in space. Are the two electrons in the same state?