Naively it seems to me that since my browser can interpret a web page using JavaScript without any a priori information, Perl should be able to as well. Is this possible? If not, why not?

JavaScript has access to a ton of things implemented in the browser, like the HTML document's DOM, various JavaScript APIs, and so on. To run JS code correctly, Perl would need to provide all of those, essentially re-implementing a browser, which is of course incredibly complex. See also the "JavaScript" section in WWW::Mechanize::FAQ.

(For the general case of running JS from Perl, there was a talk in Riga: Embedding JavaScript in Perl.)