I remember reading this in an anthology over 15 years ago:
A married couple lives in an automated house/citadel that satisfies their every need. Somehow, it also preserves them; life is so stable and there's so little need for them to do anything or think that time slows down for them, and they become gradually less responsive to anything. After much time, for some reason other people start evacuating the earth, and try numerous times to contact the couple, but they either they don't respond to the attempted contact in time, or are confident they can survive inside their self-powered house. Everyone else leaves, and there is a very long silence. The earth goes into a new ice age, but they stay safe and warm inside, barely moving as the centuries pass. After many ages pass, there's some powered noise outside, and several knocks on the door; some last attempt from a space traveler to get their attention, but after much waiting without response, the visitor leaves; much later one of the insiders asks the other something like "did you hear that?" but they soon forget it and lapse again into lethargy. After millions of years, even the atmosphere freezes, and eventually either the pressure causes the house to explode or the sun goes nova, and they die instantly.
From the tone I'd guess it's late 1950's or 1960s.