Mr. Hirway is a singer, composer and host of the podcast “Song Exploder,” which explores the origin story of a single song (Iggy Pop’s “American Valhalla,” Wilco’s “Magnetized”) with its creator. “Hrishi’s a lot younger and cooler than I,” Mr. Malina said.

The two hosts have set an ambitious goal for themselves: discuss all 154 episodes of the critically acclaimed, Emmy Award-winning series. The podcast has already gotten over 2.5 million downloads; listeners include fans who watched the series during its 1999 to 2006 run, as well as those who downloaded or streamed it later.

At one episode per week, Mr. Malina and Mr. Hirway will finish up sometime around 2019, assuming they don’t include too many special episodes — or flame out.

And then there are the guests. In addition to former cast members, the podcast has hosted Eli Attie, who was the chief speechwriter for former Vice President Al Gore, and Ronald Klain, a former chief of staff to Vice President Joseph R. Biden Jr. “I was geeking out about talking to Ron Klain, and all of that was because of ‘The West Wing,’” Mr. Hirway said.

The pair connected in 2001 when Mr. Hirway emailed Mr. Malina, a fellow Yalie, out of the blue, asking for advice about the industry. After a TV project they had been working on for two years fell through, Mr. Hirway wanted to do something more D.I.Y. Why not a podcast about “The West Wing”? It was his favorite show, his friend had been on it, and he knew about podcasting thanks to “Song Exploder.”