This episode starts off in the kitchen with the song "Maybe" from The Ink Spots playing in the background. This, along with many other 1950's songs, has been used in the Fallout video game franchise which is also set in a post-apocalyptic setting.
After Carol pours the water into the pot she places it in the stove. She then turns the knob and the burner ignites. Since modern gas stoves do jot use pilot lights they require a source of electricity in order to make the spark for the burners. Since it's well established there isn't any electricity unless they are using a generator, which they are not, then she would have had to use a match or a lighter of some kind in order to light the burner.