As Nemo said, there really isn't a ton of difference between a left-wing libertarian socialist, and a right-wing libertarian besides the fact that right-wing libertarians are against major hierarchical systems in government, and libertarian socialists are against hierarchical systems in both government and in the economy.

So no, not an oxymoron.

Are these supposed to be political oxymorons?

The 2nd is probably the most common position for people under 40 in the US.

Centralising the means of production under the state seems to increase hierarchy - instead of having many businesses in many private hands you'll get larger state owned monopolies.

Is there a real life example of libertarian-socialism in action?

I woudn't say that that young people are Libertarian, quite the opposite they fail to see the downsides of government.

That's because many people incorrectly identify libertarianism as being synonymous with neo-liberal/laissez-faire economics, but it isn't. It is doctrine which advocates policies which enhance freedoms for the most people in a given society. That is why many libertarians advocate social aspects within economies, rather than laissez-faire economics, which ultimately lead to corporate ownership, advantages over small scale operations, and advantages toward  inheriting parties leading to an economically oppressed working class.

Historically, Libertarianism began as a socialist platform to combat unfairness among the Noble class.

William Godwin, one of the fathers of libertarianism, was also one of history's strongest advocates for the estate tax. The purpose of the estate tax was to hamstring the economic advantage of the Noble upper class by taxing their estate when they transfered wealth to heirs - usually by death inheretence. The money would be transfered to the state in order to strengthen the infrastructure for the lower classes. He argued that this would create a happier and more productive society, as those who had the aptitude to rise would rise, and those who didn't would still have an easier time on their end. Meanwhile, the incompetent heirs of wealthy families would have far less of an opportunity to be a cancer on society.

Libertarian socialists don't believe the state should control means of production, they believe that workers should. That's literally what makes it libertarian socialism. There are different degrees of libertarian socialism of course, as their is with any political ideology, but the core tennants are proletariat ownership of the means of production and little government control.

However that is not libertarian though thats socialist?

Pretty much in the end supposed socialst libertarianism requires a strong powerful government. 

Pretty much I see a lot of theory and no real world applications

Firstly to invoke socialist libertarianism would require a strong powerful government to break the current status quo. 

The idea is to make every company a cooperative

In everything I read about it, the writer tries to explain why it's not an oxymoron like a cat trying to catch their own tail. Yes. it very much is an oxymoron. Libertarianism and socialism at their cores are like water and oil. "Libertarian socialism" is just an attempt to co-opt both libertarianism and socialism.

I'm curious, have you read VGPolyglot's and specially nemo37's posts on this topic here? If so, would you care to elaborate as to how they're like cats trying to catch their own tails?

The term makes sense, has been long used and is still used to this day in many circles. It just so happens that "libertarian" has been in large part appropriated by some sectors of the right, specially in some countries. It was not always like this, though, and the term came into usage as a term from the left - which also happened to be called socialist, but more on that later. Murray Rothbard, in his "The Betrayal of the American Right, p. 83, says: "One gratifying aspect of our rise to some prominence is that, for the first time in my memory, we, “our side,” had captured a crucial word from the enemy . . . “Libertari­ans” . . . had long been simply a polite word for left-wing anarchists, that is for anti-private property anarchists, either of the communist or syndicalist variety."

Moreover, it's kinda silly to argue anarchists, for example, would be incurring in contradiction by calling themselves libertarian just cause a right wing ideology took the term to identify themselves after the fact, specially when it could easilly be argued the term is representative of the former group in more senses then it is of the later.

Also, socialism has been identified with either state-economy dictatorships à la URSS, or with wellfare states. The former is just convinient (both for people pointing fingers at it as, back then, for the URSS themselves), while the later could realistically at most be said to have some socialist tendencies. What they both have in commom, though, is that most of the sectors which comprised (and still comprise) the left would disagree that they and they only are socialists.

Futhermore, the broader, older, definition of socialism is useful, as it shows what many ideiologies have in commom, from what we'd call Socialism today, to democratic currents which defend a people's government before "true communism" (aka a democratic socialism), to free-communists/anarcho-communists, to all other kinds of anarchists (minus ancap's,  but that's another discussion). It's also still in vogue, even though a lot of stigma has been built over the term. The adjective has meaning beside the proper noum. You can be "democratic" and not like North Korea's dictatorship. There's no contradiction there.

All in all, neither the words taken separate have opposing meanings, nor the term-as-a-whole is contradictory. People do believe liberty should and can - and many would argue can only - be had in a society withouth private property/wage labour/captalism/things-of-the-like-for-example. You might disagree, but that's beside the point here. If people think like that than libertarian socialist is an apt description of their beliefs. Morevor, the term-as-is, and not the simple juxtaposition of the words, has both an history and a meaning of it's own, which also makes sense and, yes, is still alive in the world, even if fringe.

I will give you though that this usage is not current in most circles, and that in those it's bound to raise questionsd/ anor cause confusion. That by no means makes the term an oxymoron.

Sorry for typos and bad structure, by the way. I should get some sleep :p