The phrase "barking mad" is a British colloquialism meaning completely insane or crazy. 

The exact origin is somewhat uncertain, but there are a couple of theories: 

One theory suggests that the phrase originates from the town of Barking in East London. There was an asylum for the insane located there during the medieval period, which led to the association of the town's name with madness. 

Another theory is that it relates to the image of a mad dog barking uncontrollably, reinforcing the idea of someone being wildly irrational or out of control.