In a libertarian utopia, most families take care of themselves by working, saving, and purchasing insurance. Taxes are low, but charitable contributions are high, and most people who cannot take care of themselves are served by charities. As James Bartholomew points out in "The Welfare State State We're In," private charities have many advantages over government programs. Finally, if people slip through the cracks of charity, government programs could be a last resort.
The part I don't really understand is why this must be a "utopia". Because something of the above is already rooted in America, and would be all the more given more honest appraisal of just how incompetent government is in managing most areas outside of public goods (i.e., national security, disease prevention, roads anything that equally benefits all citizens).