There is often this confusion between independence and libertarianism. I find that even many libertarians have confusion here. Libertarians tend to be individualistic and that is fine. But there is a perception among many (again, libertarians included) that being libertarian means isolating yourself from others.
Libertarianism to me is quite the opposite. I do not want to isolate myself from the rest of the world. I want to interact with the people around me (although not all of them). I want the benefits of mutual exchange, whether it is money being exchanged, products or services being exchanged, ideas being exchanged, friendship being exchanged, etc.
Human beings are only able to prosper through mutual cooperation and the division of labor. Being a libertarian means that I do not advocate the initiation of force for political or social change. I advocate peaceful interaction between individuals and groups of people. However, if someone wishes to be left alone and even isolated from society, it is his right to do so, as long as he is not infringing on others.
Out of billions of people on the planet, there are only a few people who would actually be able to survive while completely isolated. It would mean catching your own food, building your own shelter, and making your own clothes (if you have any). While a few people could survive, it would be a miserable life, with little in the way of luxuries.
We have a relatively prosperous society today because we benefit from human cooperation. The division of labor allows for high production and advanced trading. We don't have to catch our own food or build our own shelter or make our own clothes. We can buy these things with money, which we obtain through serving others. We have so many luxuries that we take for granted and it is because of peaceful exchange and the high division of labor.
While I understand the sentiment of libertarians who say they would like to move to an isolated island or even an island with just other liberty-minded people, I'm not sure why they would consider themselves better off. Perhaps they would be more free in their own mind, but their standard of living would be far lower, unless they continued to exchange with the rest of the world.
Libertarians are highly dependent on the people around them, just like everyone else. There is a big difference between dependence on the government and dependence on individuals around you in which you exchange goods, services, ideas, etc. Everyone is highly dependent on the free market. Without the free marketplace in which we exchange things, we would all be living at a subsistence level, if living at all.
This is why libertarians should not run away from society. I understand running away from a tyrannical country or a high-tax state, but you should not seek to isolate yourself from society as a whole. You should still try to trade and interact with those around you, even if they are not as pro-liberty minded. You can benefit from them and you may even show them the benefits of peaceful exchange.