title - What does "Zero Dark Thirty" mean?

It's humourous military slang for the small hours of the night, when you'd rather be asleep than on duty.

In addition to zero dark thirty, another term is zero dark hundred, or more commonly oh dark hundred, which, according to a qualified paratrooper (Army, Airborne) and former soldier (MI, Armor, Engineer):

In military (US) slang that period [between midnight and dawn] is referred to as
"oh-dark hundred" or sometimes "zero-dark
hundred". On the 24-hour clock the hours before
10 am start with a 0; so 1:00 am is 0100 and said
as oh-one-hundred and so forth. Thus oh-dark
hundred is anytime after midnight while it is still dark:

"The woke us up at oh-dark-hundred and ran us
thru the obstacle course." Meaning they woke us
up in the wee hours of the morning before
daylight.