I grew up in Noe Valley and remember hearing that Irish Mission District accent. The mother of one of my friends talked like she was from Brooklyn, but was a born-and-raised San Franciscan. Her 3 daughters all have the accent to a lesser degree.
I have heard a very similar accent in, of all places, New Orleans. I was told that it is common among native New Orleanians; albeit the old timers. I think it is dying out there too.