Lifeboat Foundation

Assuming the “grey goo” scenario is inevitable, how do you know that someone won’t bring a stray replicator on a space colony just before they start devouring the earth?

On its own, grey goo could not escape Earth’s gravity field, so in that respect a space colony would be safer. However, strict security measures to check people for nano-built weapons and self-replicators before they board space colonies would be essential.
 
By the way, this is a good reason to launch lifeboats as early as possible. It would be easier to search a person for grey goo than to search a biosphere — but easier is not the same as easy!
 

How do you know a nanobot won’t escape into space and devour a space colony?

Grey goo would not have the ability to launch itself into space. In theory, a meteor strike could blast goo into space — Mars rocks have been found on Earth. For a variety of reasons, it would be best if the space colonies were placed as far away from Earth as possible.
 

Is it really inevitable? Nanobots need some kind of fuel.

Life has covered the Earth (including the atmosphere and lithosphere) simply by using available “fuel sources”. Nanotechnology-based machines could do the same.
 

Why build a replicator if its only function is to build more replicators? How will that help anyone do anything?

We have to worry about hobbyists building useless goo just because they can. We also need to worry about destructive scenarios such as nano-fueled arms races and war.
 

According to a NASA study, a colony needs a population in the hundreds of thousands to support any kind of industry. How could we get that many people to live in a space colony? It would be very expensive.

This study must have been based on out-of-date technology. In theory, all that is required is human stem cells and robots to create the human colony at the desired destination (such a method could be used for interstellar space travel, for example). Our plans for space colonies, however, contain 1,000 fully developed humans.