If you're talking calendar date (e.g. "12th of December"), we can't generally be confident about dates in antiquity unless there are astronomical events to calibrate the calendar of the time. There's no real certainty in any ancient calendar dates until the institution of the Julian calendar in 46 BCE, in the Roman world, and the Han calendar in China. There's not much to help us synchronise other ancient lunar calendars with the Julian or Gregorian calendars because lunar calendars tend to deal with the problem of intercalation very erratically; it's only once you get a solar calendar (the Julian, and a few others that were developed later but have since been abandoned) or a very formalised lunisolar calendar (the Han) that we can be confident about specific dates. When Cicero writes a letter to his friend Atticus in 49 BCE dated "the 2nd of February" (or in Latin a. d. IIII Id. Febr.), the margin of error on the actual date is anywhere up to half a season. But when he writes a letter in 44 BCE dated to 15 April, we can be confident of the exact day because the Julian calendar had been adopted at that point.

If you're talking calendar era (e.g. "2014 CE"), then that matters a lot more (who really cares what exact day an event happened? the year is much more important), but very laborious. The answer is: with great difficulty, and over a looooong period of time. It's a ridiculously fiddly process and relies on evidence from all over the place. That doesn't necessarily imply any uncertainty in the results: corroboration is sought when that's possible, and historians of antiquity will (if they're being conscientious) indicate the areas of remaining uncertainty.

The most useful tool is a genre of historical document known as "chronographies". These are basically just works that give a huge long list of dates and major events that happened that year. The Byzantine documentary record is particularly rich with these. We can often be very confident about the older events that they list because we know quite a lot about the work of earlier chronographers, even though their own works don't survive.

For Classical-era Greece, the extant chronographies owe a lot to the chronographic work of Eratosthenes in the 3rd century BCE. He did an awful lot of legwork in synthesising the haphazard chronographic information he found in older historians; and he also adopted the tremendously important innovation of tying calendar era dates to a single numeric count (the Olympiad count). Most calendar era systems at the time used eponymous years, e.g. "the year that So-and-so was archon at Athens", so any time you wanted to work out which year something happened, you had to have huge long lists of Athenian archons, Spartan kings, Olympic victors, and Argive priestesses by your side, depending on which city's history you were looking at. With the Olympiad count, everything was reduced to a couple of numbers counting from a specific date (which corresponds to 776 BCE, as it happens; so for example we know from later chronographies that Eratosthenes was born in the 126th Olympiad, i.e. 276-272 BCE [not 276-273; the Greek calendar started in midsummer, so the first year of the 126th Olympiad was 276/5, the fourth year was 273/2]). This combo made his work especially important. He wasn't the first to use either innovation: the idea of counting in Olympiads was first developed by Timaios of Tauromenion, some decades earlier, and several historians had previously been working on synthesising chronologies using the horrible old eponymous systems. The earliest of them was Hellanikos of Lesbos, roughly contemporary with Herodotos: he based his chronology on the eponymous system of the priestesses of Hera at Argos.

In other locations, other chronographic efforts applied. Egypt used an eponymous system based on how many years so-and-so had been pharaoh, and the pharaoh lists go back a very long way (though there are increasing uncertainties the further back you go). Rome used an eponymous system based on who was consul in a given year (and again, with uncertainties for all dates in the 4th century BCE and earlier). Assyria had its own eponymous system, the Han calendar started a new calendar era every five years or so, and so on and so on. By collating all of these together and matching them up with a single system -- the BCE/CE system -- and trying to sort out inconsistencies and errors as you go, you eventually arrive at a pretty good unified chronology. Uncertainties within each place's eponymous system will be reflected as uncertainties in the BCE/CE date.

Edit. Corrected year numbering error.