They absolutely weren't socialists. Here's a free tip: Jonah Goldberg is a political hack whose career is dedicated to smearing liberal ideas and individuals. He's not a historical scholar and has been previously caught fudging elements of his resume. He's not a good source for arguing that the Nazis were socialists.
The "case" that Nazis are socialists is mainly based on two things. First is that their full name is National Socialist German Worker's Party. This is a phenomenally stupid argument, however. If we use this to believe that the Nazis were socialists then we would also have to believe that North Korea, one of the most brutal and repressive dictatorships in the world, is a democracy. After all their full name is the Democratic People's Republic of Korea.
The second main argument is that the Nazis favored a strong state, including a robust social safety net (at least for Aryans). But this argument, while not as stupid as the name argument, is also nonsense. Acceptance of the state has nothing to do with left-right alignment and you have people on both sides who laud the state and reject it. Supporting a social safety net also does not make one socialist. Governments of all stripes, including conservative ones such as Reaganite America and Thatcher's Britain, have supported a social safety net. In fact, the modern social safety net was created in 19th Century Europe by conservative politicians.
What makes one a socialist is acceptance of beliefs which constitute socialism. And the Nazis fundamentally rejected all of these ideas. Most fundamentally they rejected the idea, central to socialism, that the fundamental dividing line human society is class. That's the bedrock of socialism and the Nazis completely rejected it. For the Nazis the fundamental diving line of human society was race. The Nazis also rejected central socialist ideas like dialectic materialism and the eventual establishment of a utopian "post historical" society.