How would a minarchy be established? Would it be democratic?

A government not supported by the majority cannot sustain itself without tyranny. They will eventually get rid of the current government and laws, even amend the constitution, if needed, to get their way. Thus it is impossible to maintain libertarianism by restricting voters or their power.

To achieve a libertarian country, either the majority of people in an existing country must be convinced of its merits, or libertarians must come together to form a new nation. And if the majority is libertarian, they will vote as such, maintaining the minarchy.