On many episodes of Star Trek DS9, each character has a good sense of time. Whether it's on a mission or having a drink at Quarks, they seem to know when to begin that mission or have their social gatherings. My question is, how is time measured in space when they are nowhere near Earth? They can't use Earth time in particular, since, for one thing, that time is based on the rotation of the Earth around the Sun. Anyone care to elaborate?
Starships typically use an Earth Day (presumably a relic of Earth being the major founding partner of Starfleet/The Federation), and it's no stretch of the imagination to assume that starships do too. They also have clearly defined night shifts (Beverly Crusher is mentioned as occasionally taking a night shift to keep her command skills up, and Harry Kim is seen to take them in Voyager).
DS9 presumably also follows this, although may use the 26 hour Bajoran day, since it's actually a Bajoran space station located in the Bajoran system.
Other than that, time is generally measured in Hours/minutes/seconds and days, and the Stardate. A month is mostly forgotten about.
The general assumption (although not, as far as I'm aware, verified) is that actually all characters will refer to "Days, hours, years" etc in their own native tongue, along with measurements and similar, and the Universal Translator will translate units at the same time.
eg if an alien says he's 70 years old, he may actually be giving his age in his own language/frame of reference (200 cycles, 20,000 days on his planet, or halfway through his second metamorphosis), but the UT will know that those numbers are the equivalent of 70 earth years, and will translate it to that to give the listener a frame of reference they can understand.