Hawking points out what others have been saying for some time. We cannot afford to have all of humanity on one globe and just hope for the best. Plague, war, famine, asteroids, ice age and what have you, could wipe out all known sentient life in very short order. Either we grow or we die. The big question after that is how. The biggest problems are food and water. Air can be recycled using existing technology. Water is more of a problem. Treatment facilities for water don't clean it as much as it should. You'd need some sort of biological component to process the waste materials. Food is a major headache as well. People need lots and lots of it and self sustaining farming on an orbital platform or ship would be daunting to say the least. In the past when scientists have tried experiments with biospheres, that has been a major chokepoint. In some cases, normiall mild mannered scientists have had physical fights over food allotments. That does not bode well for ventures where one cannot simply leave if there isn't enough food or if you're tired of just getting by.
The scale of such an enterprise is staggering. Think of the materials you'd have to get into space and then assemble before you could go anywhere. Given that our current lift capacity for orbital and suborbitals is so limited, the cost of such a venture would require the combined might of many nations. That alone, makes it a non-starter. We cannot agree on other things that are relatively simple by comparison. Imagine the politicking and graft that would surround such huge expenditures. Even with the advent of the space elevator becomes a reality we will not be able to put aside petty squabbling until either a handful of very dedicated people with nearly inexhaustible resources starts or we are really and truly threatened with extinction.