The problem with building a spaceship is deciding what to take with you. This is particularly true if you are leaving the Earth to escape a catastrophy. For example, if an asteroid is going to hit the planet, who do you take, and who do you leave behind? What might you forget to take that would be vital to your survival?
I propose that the entire planet should be the spaceship. Obviously we can't put rocket motors on the Earth and move it because of the Earth's atmosphere, but we are extremely fortunate to have a large airless Moon. We could place motors on the Moon and use these to move the earth.
In figure 1, you can see the thrust vector being applied to the Moon at two points in it's orbit. The first of these impulses would increase the orbital velocity causing the moon to stay on that side of the Earth for a longer period than normal. The second impulse would restore it to it's normal orbital path for the second half of it's orbit.
If we were in a hurry and wanted to move the Earth with current technology, we could use crude nuclear explosions for propulsion by covering suplus nuclear weapons with lunar dust and exploding them. This technology could move the Earth 1/2 it's diameter in about 30 years. Unfortunately at the end of that time, we would be out of weapons ( maybe a good thing ) and out of dust.
A more sustainable approach would be to use ion propulsion. This would have the benefit of using much less of the lunar dust for propellant. It would require power stations to be built on the moon. These stations could be solar power, but probably should be nuclear power.