The first thing the survivors should do is find any large body of water, preferably the sea and as far away from the blast site as possible. According to the Centers for Disease Control and Prevention (CDC), one can decontaminate oneself in water as the water can “dilute radioactive material to very low levels”. Given that there was no rate at how fast the radiation was spreading, the lucky survivors would have had enough time to think and talk among themselves.
If they are all to be saved, it would be impractical to build a spaceship that would accommodate all of them. A better option would be to use the spaceship as a shelter and wait out the radiation. They could stock up on food and water supplies. Sure enough, the spaceship is spacious, it just cannot fly 12 people and a cow. Perhaps some of them may know how to lessen the amount of radiation exposure in their body, certain food or chemicals they can ingest, or how to lessen contamination in the water. They can start gathering food and water as soon as possible. Store-bought water and canned food would be best to stock up on if they are near a city, or harvest food and water that is available to them out in the wild and look for naturally-occurring “decontaminators”. Being the only survivors, they can use all resources available to them.
The spaceship can be sealed off with its cabin pressure regularized to replicate conditions much like on Earth solving the problem of finding safe air to breathe. They can come up with an air purification system that draws harmful air from the outside and convert it to breathable air. Given it is a spaceship, there must be an oxygen supply tank and backups they can use for a few weeks. In addition, they can also come up with an effective water harvester and purification system using the nearby large body of water as a source.
They can utilize hydroponic farming (soilless farming) once they have their basic water needs taken care of. This solves their food supply. Space suits located in the cabin of the spaceship can be used to venture out in the open as the radiation spreads overtime. These will be helpful in gathering more supplies or finding other potential survivors.
Should all go well in the end, the survivors would have successfully waited out the radiation and Earth would be habitable again. The survivors can start looking for arable land and more bodies of water. Eventually the survivors can start repopulating.
However, what led to this catastrophe in the first place?
Conflict of interest. Humans have resorted to nuclear warfare because a disagreement between parties arose. A certain party thought nuclear warfare was the answer and set off a large explosion that almost wiped out the entire human race.
What could be done to mitigate such a disaster?
Survival shelters are a good way to survive a nuclear explosion. At a minimum, the shelter must have a reliable food and water supply and air purification system. The shelter must be strong enough to withstand at least the shockwave generated by the explosion. The shelter must be accessible and safe as well. It may be expensive to make (or buy) a survival shelter, but the investment is worth it given the fact that a nuclear explosion is sure to happen.
Would it have been possible to avert a nuclear explosion? What could can have been done to avoid a nuclear explosion?
A good preventive measure would be to educate children, at a young age, that violence is never the answer, that they should settle an argument by reaching a consensus. Furthermore, it helps to teach children of reasonable age a bit of world history about the catastrophic results of a nuclear explosion and their effects. A lot can be learned from the bombing of Hiroshima in World War II, for example.
In the long run, simple education of how to deal with conflict, and the effects of bad decisions made by political leaders would create better world leaders capable of resolving any conflict that would not inevitably lead to nuclear warfare, or at least create leaders that care about the masses.