I seem to post stuff like this fairly frequently in space travel-related stories, so I thought I'd just write it all up here. Summary - Yes, it would probably be a nice thing to have a truly self-sufficient human civilization on another planet so that we are insulated against any catastrophe that might happen to the Earth, but such a colony is basically not possible to create now.
If you're going to be self-sufficient and a backup against the Earth being wiped out, then you must be able to manufacture everything your colony needs in your colony. No country on Earth currently does this now. If one of the continent-spanning countries like the US or China or the EU as a whole suddenly had to exist with no trade at all with any other country, then it would probably be tough for a while, but they could do it. Your colony would have to have a comparable size and level of economic activity for it to be possible.
Let's take IC fabs as an example. If your colony will depend on computers in any way, then you'll have to have one. A huge operation, incredibly expensive even on Earth, demanding lots of specialized personnel. Can you imagine building one on another planet right now? How about building several of them, and having the materials and knowledge base to build more if needed? And that's just one of the highly complex manufacturing facilities you'll need. Steel foundries, aluminum, other metals, casting, forging, pressing, something with hydrocarbons for making plastics, something like concrete, stuff for growing plants and raising animals, thousands of other things that I haven't thought of yet. You would need to move a major country's worth of people, expertise, and capital to your colony planet. You can't just rough it and go without on a planet with no atmosphere.
Right now, we're at the level of moving one robot about the size of a car to Mars. We've got a hell of a long way to go before we can think about setting up even a dinky just-because-we-can colony with a few dozen people that is dependent on the Earth for critical supplies, much less anything that can hope to survive the loss of the Earth. I believe that we'll get there eventually, but it's not something that's even remotely on the horizon right now.
Another way to put it - there is no imaginable catastrophe that could make the Earth less habitable than, say, Mars already is. If we have the technology and the will to set up a real colony there, then we also have the technology and the will to survive whatever might happen to the Earth, and for a lot less expense.
And we haven't even gotten to the type of government and lifestyle that such a colony would need. If you're going to be hard up against the lower size limits, then anything like first-world freedom is probably impossible. If there aren't enough resources to support much of a non-productive population, then anyone too old or disabled to do useful work would have to be sent back to Earth or killed. Reproduction would have to be tightly controlled - too many or too few children at any time could be a disaster, if it either would raise the population beyond what the infrastructure could support, or would not lead to enough workers to maintain the infrastructure or preserve key knowledge. You'd probably need a policy like every woman must bear exactly 2 children before she turns 25, and no more, ever. Education and careers too - you don't want to be a welder, say? Too bad, the colony must have a welder and you've been chosen. Get to work, or else. Getting out of all of that would either require a much larger colony or frequent trade with Earth. But there's not much point in creating an in-case-of-emergency colony that depends on trade with Earth to maintain its lifestyle, so you'd have to go with size.