The Austrian school’s insights on economics did not always imply that there is no room whatsoever for governmental intervention. This was not the view taken by Mises himself.
Oftentimes, people unfamiliar with the Austrian school tend to bundle it with political anarchism. There are—I believe—different possible explanations for that, but the most prominent ones seem to me the following two. First, it’s true that some Austrians can in effect be considered political anarchists as well. Second, the Austrian school is not only a school of thought about economics, but also (at least) about epistemology and political philosophy—i.e., it’s concerned with issues such as the relationship between individuals and states.
However, all Austrians share a common denominator: they all accept the teachings of Human Action, Mises’s magnum opus. Hence, in order to determine whether Austrianism does necessarily imply political anarchism as well, it might be sensible to scrutinize Human Action: What does it say about the role of state and government in society and how far they can go in subjecting individuals?
Is Unfettered Individual Freedom a Natural Right?
Many anarchists ground their anarchism on the concept of “natural law” or “natural rights”—namely, the idea that something as “natural law” does really exist and that it mandates the inalienability of individual freedom to the government or the sovereign.
However, such a stance entails two logical problems. First, one needs to postulate the existence of enforceable rights which do not coincide with the ones established by law. Second, one needs to compellingly conclude that such rights are objectively in favor of freedom rather than some other ideal—say, equality, tribalism, nationalism, etc.
On the first topic Mises’s answer is straightforward enough: justice exists only insofar as it is established by law. As a matter of fact, he writes,
The notion of justice can logically only be resorted to de lege lata [i.e., the law as it exists]. It makes sense only when approving or disapproving concrete conduct from the point of view of the valid laws of the country….There is no such thing as an absolute notion of justice not referring to a definite system of social organization. (Human Action,  1998, p. 717, emphasis added)
Moreover, Mises does not only agree on ultimately resorting to laws in order to asses justice, but he goes even further—embracing a viewpoint about freedom and, more generally, human cooperation, rooted in a contractualistic philosophy. In fact, Mises writes,
It is therefore nonsense to rant about an alleged “natural” and “inborn” freedom which people are supposed to have enjoyed in the ages preceding the emergence of social bonds. Man was not created free; what freedom he may possess has been given to him by society….Liberty and freedom are the conditions of man within a contractual society. (Human Action,  1998, p. 280, emphasis added)
On the second topic Mises is manifestly skeptical when it comes to hypothesizing that we can find, or prove, any kind of objectivity about “natural laws”—or about “natural morals” that we can derive cogent “natural laws” from. In fact, he writes that
There is, however, no such thing as natural law and a perennial standard of what is just and what is unjust. Nature is alien to the idea of right and wrong. “Thou shalt not kill” is certainly not part of natural law….The notion of right and wrong is a human device. (Human Action,  1998, p. 716, emphasis added)
Furthermore, Mises goes as far as stating that, were a “natural law” to really exist, it would be sensible to think of it as a ruthless law of abuse and aggression—in accordance with which no social cooperation nor division of labor would be feasible. As we can read,
nature does not generate peace and good will. The characteristic mark of the “state of nature” is irreconcilable conflict. Each specimen is the rival of all other specimens. The means of subsistence are scarce and do not grant survival to all. The conflicts can never disappear. (Human Action,  1998, p. 669, emphasis added)
Lastly, Mises clearly lays out his skepticism about “natural rights” as a useful concept when it comes to defending property rights—a fallacious argument that can be refuted, on a priori grounds, by those claiming that “equality” rather than “property” is what “nature” imposes upon human beings. In Mises’s own words,
It is useless to stand upon an alleged “natural” right of individuals to own property if other people assert that the foremost “natural” right is that of income equality. Such disputes can never be settled. (Human Action,  1998, p. 281)
The Right Place for Government
Once the “natural law” argument in favor of anarchism is refuted, we are left with an open question: Is Government useful to human beings? For Mises, the answer is “yes, it is”—but with a few caveats.
Without entering into the details about how detrimental government interventionism is for the economy (part six of Human Action is entirely devoted to it, and Austrians are well aware of the damages caused by currency manipulation, trade barriers, legal monopolies, labor unions, etc.), we cannot deny that Mises conceived of government as something that, taken with a grain of salt, could foster human cooperation and prosperity.
For instance, he writes about taxes and government that
As far as the government fulfills its social functions and the taxes do not exceed the amount required for securing the smooth operation of the government apparatus, they are necessary costs and repay themselves. (Human Action,  1998, p. 738)
Lastly, Mises’s distrust of natural spontaneous social order and anarchism is clearly set out at the very beginning of part two of Human Action, where we can read:
An anarchistic society would be exposed to the mercy of every individual. Society cannot exist if the majority is not ready to hinder, by the application or threat of violent action, minorities from destroying the social order. This power is vested in the state or government. (Human Action,  1998, p. 149, emphasis added)
Therefore, we can conclude that Mises accepts indeed a role for governmental intervention, that is, the enactment and the enforcement of the rule of law—whereby “naturally” weak members of society are protected against violence and abuse on the part of stronger ones.
Contrarily to what is often claimed, one can consider him/herself an Austrian disciple even without feeling necessarily compelled to be bundled with political anarchism: Austrian sound teachings about economics do not necessarily imply that no room whatsoever is left for governmental intervention.
Nonetheless, the government’s role should be kept to the bare minimum, and its attempts to meddle with resource allocation—via fractional reserve banking, fiat money, legal monopolies, trade barriers, etc.—must be forcefully and explicitly opposed.