Assume we have two people who wish to enter into a private contract. One of the two wants to sell himself into slavery to the other. According to libertarian principles, no one has a right to interfere in that contract.
Think I'm exaggerating? This is what the Stanford Encyclopedia of Philosophy has to say about libertarianism
It wrongs an individual to subject her to non-consensual and unprovoked killing, maiming, enslavement, or forcible manipulation. (emphasis added)
Full-self ownership is sometimes thought to guarantee that the agent has a certain basic liberty of action, but this is not so. (emphasis in the original) For if the rest of the world (natural resources and artifacts) is fully (“maximally”) owned by others, one is not permitted to do anything without their consent—since that would involve the use of their property. For example, as a result of one's trespass on their land, one may become their slave. The protection that self-ownership affords is a basic protection against others doing certain things to one, but not a guarantee of liberty. (emphasis added) Even this protection, however, may be merely formal. A plausible thesis of self-ownership must allow that some rights (e.g., against imprisonment) may be lost if one violates the rights of others. Hence, if the rest of world is owned by others, then anything one does without their consent violates their property rights, and, as a result of such violations, one may lose some or all of one's rights of self-ownership. (emphasis added) This point shows that, because agents must use natural resources (occupy space, breathe air, etc.), self-ownership on its own has no substantive implications. It is only when combined with assumptions about how the rest of the world is owned (and the consequences of violating those property rights) that substantive implications follow.
Libertarianism is the philosophy of libertines and teenagers (but I repeat myself). It is incompatible with Catholic philosophy. Catholic philosophy views "ownership", in the strict sense, as an attribute of God - God owns all things because He created all things.
Libertiarianism directly contradicts the Declaration of Independence:
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.The right to freedom is "unalienable" - I am not capable of entering into a contract of slavery, on either side. But libertarianism must allow such a contract, or it isn't libertarianism.
Every human individual is, at most, a steward of one or more created goods. Strictly speaking, in Catholic philosophy, we don't own anything. Our lives revolve around the rights that flow from being appointed a steward and exercising stewardship, they do not flow from ownership. Thus, the core concept of libertarianism is fundamentally incompatible both with being an American and with being a Catholic.
Most libertarians can't even define the word.