You need to have a Space Port built in one of your cities. The ship itself needs parts, not unlike those needed in games of Civ1 and Civ2 but much harder to construct. As with other things you build, the resource cost is 25% less than standard if you are on "easy" level and 25% more on the "insane" level. The in-game manual tells you the cost for your level.
Each Colony Ship Part must be built using a specific rare metal: cobalt, mercury, or uranium, each of which, in a randomly-generated world, appears (to players who have researched Mass Production) on only two of what were formerly Dead Lands tiles.
- Cobalt appears as a pinkish-brown crystalline "flower" and is used in the Colony Ship Component, which requires Transstellar Colonization and costs 240 resources; six are needed, i.e. total cost 1440 resources
- Mercury appears as a set of three silver globules and is used in the Habitation Module, which requires Self-Contained Environment and costs 800 resources; two are needed, i.e. total cost 1600 resources
- Uranium appears as a glowing silver rod and is used in the Power Module, which requires Impulse Drive and costs 600 resources; four are needed, i.e. total cost 2400 resources
Producing each part requires the city to have one worker using the metal, normally requiring Full Control rather than any of the automatic population management options. If there is no worker there on a particular turn, the part gets no resources added to it.
Clearly no more than two players can be serious contenders in the space race on a standard map (though you can modify any map or create your own before playing). If you want to end the game as fast as possible, try controlling and using both deposits of each metal, particularly Uranium.
Hit F8 to check progress with your ship and see if it has competition.