Still a line from "The Marvelous Ms. Maisel"
The character said:
And Stan... who has a voice that bored a thousand ships into sinking themselves.
I was totally lost about this phrase.
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It is a clever pun that plays on the idea of "the face that launched a thousand ships". This phrase is a reference to the launching of a fleet of ships to rescue Helen of Troy, a woman of the most astounding beauty who eloped/was abducted by Paris. The launching of the ships to bring her back sparked the start of the Trojan war (the Iliad). It is such a poetic phrase that it is in common usage in English to describe someone of exceptional beauty ("she is beautiful enough to launch a thousand ships") . The phrase "has a voice that bored a thousand ships into sinking themselves" plays with this idea to suggest that Stan's voice has a level of boredom that is as exceptional as Helen of Troy is beautiful, and does it in a clever way. Rather than launching a thousand ships to rescue the most beautiful woman ever, this guys voice is so dull that a thousand captains would rather sink their own ships than listen to it.
It is a reference to the Iliad, in which a lady leaves her husband but she was so pretty that he was willing to launch a thousand ships full of Greek soldiers to get her back.
So you get the saying "beautiful enough to launch a thousand ships" to mean "very pretty." And now there is apparently a bloke who is so boring his voice will sink a thousand ships, that is, he is very boring.
While the underlying reference, as Quuxplusone stated, is to the Iliad, the specific expression is a play on a famous line from Christopher Marlowe's Doctor Faustus:
Was this the face that launched a thousand ships
And burnt the topless towers of Ilium?
The context is that the main character, Faust, has made a deal with the devil and gained special powers, including the ability to talk with the spirits of the dead. Here, he has summoned the spirit of Helen of Troy. He asks if this is really the spirit of a woman so beautiful that her abduction motivated her husband to launch a huge naval invasion to get her back.
It means that his voice was so boring that a thousand ships decided to sink themselves. Compare it with the expression to talk someone into doing something.
talk into (phrasal verb) If you talk a person into doing something they do not want to do, especially something wrong or stupid, you persuade them to do it.
It's basically the same idea.
(answer transcribed from comment)
As well as what the other answers have said, the sentence is a pun. The term "bore" can mean 'drill a hole in something', or can mean 'induce boredom', so the sentence has two meanings. The second is that Stan's voice makes holes in ships and causes them to sink.