Wikipedia is actually overstating how specific it's possible to get, but according to their editorial rules and it's possible to get within a set number of dates. There are two chronologies dates like these are based on: a "relative chronology"(sequence of events) and the "absolute chronology"(where those events fall in BC/CE). The internal chronology is given by a combination of documents. The most famous are the King Lists, which are (sometimes politically motivated and thus suspect, but still very useful) lists of kings along with their regnal years. In addition we can make use of the relative dates used on Babylonian and Assyrian tablets, the Year Name or Limmu(in Assyria) system. The Year Name system gave a year the name of some important event and we can use the years given by the year-name system to fix the chronological framework of our inscriptions. So using the Year Names recorded, the surviving Sargonid inscriptions, and the Sumerian King List we can come up with a quite good internal chronology of how long each king reigned and in what order.

The tricky part is pegging those chronologies to BC/CE dates. The main way we do this is with astronomical data recorded in the ancient sources most famously Babylonian astronomical diaries and for our period the Venus Tablet of Ammisaduqa(a tablet recording observations of Venus made in the 8th year of Ammisaduqa's reign in Babylon).By comparing the astronomical data in the tablet to modern back-projections of the rising and setting of Venus, we can correlate the astronomical data with the regnal years mentioned and ideally work back from there. The traditional chronology of Mesopotamia is largely based on one of three possible chronologies that would fit the Venus Tablet(a Low, Middle, and High chronology; Wikipedia's date is that given by the Low Chronology but that is a matter of standardization for them). However, there are definite questions about how reliable or fixed the Venus tablet is-if for example it is a compilation of earlier tablets, in which case that's going to throw things off-so more and more scholars are making substantial adjustments to the chronologies based on archaeological evidence, textual data, and improved radiocarbon and dendochronological dating, all of which will likely refine our understanding of Mesopotamian chronology.