Post-apocalyptic family survivors of a nuclear attack are grouped together for days in the basement of their house

I watched this (B&W) nuclear movie in 1988 on TV. Post-apocalyptic family survivors of a nuclear attack are grouped together for days in the basement of their house, where fear and dwindling supplies wear away at their dynamic. After days waiting for the radiation level to come down, the man gets out of the basement and he sees many dead bodies on the streets. The man starts a journey to find a unique place on Earth which is free of nuclear radiation.

On the journey a woman follows him but he doesn’t want her near him because gangs could kill them. At the end he found a ship in which there is food for fifty years but nobody gets into it and nobody who is inside the ship gets out. Finally, he gets to climb on board the ship and escapes swimming to the beach where the woman is waiting for him. That is the only color scene of the movie at the end on the beach with the woman.