The story
From U.S. military terminology, in which hours before 10:00 am are prefixed with a zero, read as "oh", as in 0530, oh five-thirty, for 5:30 am.
Etymology from Wiktionary, CC BY-SA 4.0.
oh dark thirty
Today it means: Some unspecified hour in the early morning (implying an unpleasant time to be awake). Full definition →