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According to the Japanese dictionary, it means "sink or swim", "do or die", or "high stakes", but I can't figure out the etymology.

The literal meaning appears to be "one or eight". So does it literally mean "the odds are 1 to 8"?

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Shogakukan's Kokugo Dai Jiten Dictionary states:

カルタ賭博から出た語
A term from card gambling

The entry also gives 一{いち}か六{ろく}か as a synonym. This version may be more obviously related to dice gambling, but as mentioned in the discussion at http://gogen-allguide.com/i/ichikabachika.html, the 一 and 八 (or 六) here may not directly indicate the numbers, and might instead be slang or jargon for odds or evens. Shogakukan's definition of "choosing one of two choices" seems to back this up, and the additional definition of "leaving one's luck to fate and bravely doing something" and additional synonym of 伸{の}るか反{そ}るか suggests a kind of "all or nothing" implication.

So ultimately, this doesn't have to do with "one to eight odds", and instead 一か八か indicates a 50-50 all-or-nothing gamble.

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