All the definitions I've seen here have a positive connotation, which is completely wrong. The phrase has always had a very definite negative connotation, and means to be supremely screwed, screwed beyond all comprehension. The original phrase was "screwed, blewed and tattooed".
"Screwed" essentially means "cheated" here, much as it does today.
"Blewed" meant "lost or been robbed of". The word's origin is from the German "blauen" so it's actually related to "blue", not "blew", and meant that something had vanished (into the blue). (According to "A Dictionary of Slang, Jargon & Cant" By Charles Godfrey Leland, published in 1889.)
"Tattooed" refers here to a beating with very rapid blows, in the same sense as a military tattoo, which is a rapid pattern on a drum.
So, the phrase literally meant "cheated, robbed and beaten".
I invested my life's savings in a fraudulent investment scam, and now I'm screwed, blued and tattooed.