The style sheet I received when I was working on texts for publishers suggested 1920s, but did not claim that 1920's was incorrect. Writers need to be aware that all magazines, newspapers, and publishers have their own style sheet for many of these questions and that there is no unanimity among them. BTW, the answer above that 20's is correct because the apostrophe indicates something missing as in a contraction is incorrect. One of the reasons apostrophe use is so confusing is because using 's to indicate the plural form of things like decades, or grades on report cards has always been acceptable. It is this rule that confuses writers who then incorrectly use the 's for plurals such as apple's and other nouns such as family names (the Smith's). It is to avoid this error that style sheets recommend the lower case s as the plural for I received 4 Bs on my report card, but if I am an A student, it is context that explains that As stands for grades not the comparison.
Best solution in such situations--find an alternative. Write the Roaring Twenties.