Firstly, I must admit that I disagree with Rothbard and Block and Hoppe on some things. If you look through my post history you'll find that I never blindly direct people to read these authors, and will argue the philosophical, political, legal, or pragmatic issues with my own thoughts and deductions.
Second, there is no contradiction in Rothbard here: he very clearly states that no one owns their reputation, but that a "reputation" is an opinion that someone else owns about you. You can certainly try to improve your reputation and defend it from attacks, but when he says that there is no legal basis for libel he means that person A's spreading of information about B is either true and should rightly be disseminated, or it is false and person B can demonstrate it's falsity, or person A will quickly erode their own reputation for honesty. Especially today when information is cheap and widely disseminated there would be little risk of libel permanently effecting one's earning potential. And even if it could, why should a business owner expect a communally supported legal system to defend his from these charges? Couldn't "Libel Insurance" develop to protect income streams and work to counteract untrue attacks?
This brings me to my third and central point, that in a anarchic or volutaryist or ancap system you'd be free to join a legal association/govt that forbids their members from disseminating libel, and seeks to justly recover from those non-members who libel the members. They would only be restricted by what the body of non-members found to be unjust - and they would quickly reach an equilibrium that represented the point at which the quality and cost of protection against libel truly reflected what their members were willing to pay for, and at which non-members were satisfied that the rules were being applied justly.
Rothbard fleetingly wrote what he thought the "perfect libertarian law" should be. He had some good insights, and some things that I don't think are right. But that shouldn't stop you from embracing your own reasoned version of a political philosophy (and if you've followed discussions on mises.org, or here, or elsewhere, you'd know that Rothbard is venerated, but only a few people take what he wrote as infallible dogmatic gospel). I wouldn't let Hillary Clinton's musing about the role of a state dissuade me from considering other arguments in favor of a state, so I ask that you don't say "You see, this one guy wasn't perfect, therefore you're all wrong."
I hope that didn't just sound like crickets to you.