No, it does not seem to be the same story as The machine stops.
My guess, after watching episode 2 is that the "outside" we've seen so far is not any outside at all, but just a chamber filled with poison. It was very obvious, of course, that the helmet was a screen, but that just made me think the outside maybe was really poisonous. I feel a bit stupid that I didn't think of the second layer of the Matrix until computer guy mentioned the door at the end.