It's one half of a phone call from Dragnet season three, episode five- "Homicide/DR-06" which aired on October 24, 1968.
At first, I suspected a continuity error, but Mad Men season six is shuffling through 1968 expeditiously.
I call this episode "Joe's Apartment" because Homicide/DR-06 is a clunky title. This episode is a bookend to "Let's go home with Officer Gannon" - (season two, episode five/"The Big Neighbor").
HOLD THE PHONE
Start streaming Dragnet at about twelve minutes and you get the exact clip that was used on Mad Men.
It even has a cute sequence of Joe bringing steaks by Bill on his way to the hibachi that is pure "Bert & Ernie."
If you have been watching what Don Draper and the other characters have been watching and wondered when the channel would change to Dragnet, this is your episode.
Other shows featured on Mad Men in the background include The Man from U.N.C.L.E., The Donna Reed Show, The Three Stooges, The Patty Duke Show, Bewitched, Hawaii Five-0, People Are Funny, and more. The characters are also seen watching a lot of the news and various assassinations and riots per the historical timeline.