Uhh, not quite.
This is how you do it:
"My best friend said, 'Like, yeah, and so he said, "Quote marks are dumb," like, so he went on and said, "my professor said that, 'John Barth was a bane on editors everywhere.' he said." she said,' my best friend said."
Got it? It's basically alternating " and ' (double and single) quotes. Go ahead and read some of John Barth's stuff, which uses this all the time, for some examples.