The Core
What Would Happen When The Core Freezes?
Diagram Of The Earth's Inner Layers
The Core of the Earth | 100 Greatest Discoveries
What Would Happen If The Core Froze?
Less Earthquake & Volcanoes
That's great. But soon the wind and rain will erode away the Earth.
Mountains Will Shrink
Get washed away by rain and rivers, into the sea. But will take a long time to happen.
Bad News In The End
There will be no more land left! No continents, no rocks, no islands, nor sand banks. All that will be gone forever.
Naked Science - Earth's Core
No More Land, So What Else?
Even though it will take awhile for the land to break down and be gone, you'll still be able to live on the ocean with a ship of some sort. That is until the magnetic field weakens, and is fully gone.
Soon the deeper liquid metal core starts to freeze up. That would result in both the magnetic field, and water dissolving. But most importantly, all air would be lost.
No Atmosphere
Radiation everywhere! Which would result in multiple cases of cancer and blackouts.
No Water
With no atmosphere, there is no air pressure to hold down the water. All water would vanish. So, the ship might not work after the ocean turns into a lake, then into a pond, then lastly into nothing.
No Oxygen
No atmosphere means no oxygen. Thats a given. Unless you have a astronaut suit that keeps generating oxygen into the suit, you're a goner.
The Next Mars?
Mars had gone through this process a few billion years ago. Because it is smaller than Earth, it froze up sooner. Earth is believed to go through this process of core freezing, and ending of humanity. But not anytime soon.