You'll notice the map for the Persian Empire has varying levels of details, and the largest part of that comes from Herodotus. Although he doesn't give exact geographic boundaries for the satrapies (assuming there were), he does list the peoples who existed under the authority of the satrap, and from thus we can make pretty good guesses based on reconstructing the archaeological evidence of the extent of those peoples. It also helps that the vast majority of these satrapies were not artificial constructs, but rather, the imposition of Persian will over existing polities or groups of polities. This means that, knowing the extent of those polities prior to their incorporation into the Empire, we can make reasonable guesses as to the borders of them. Whether Darius or Xerxes had maps detailing the exact borders of these polities is unclear, although it was probably unnecessary given that tax burdens fell on immovable polities and their environs, and there largely shouldn't exist peoples that feel between the cracks.

Persia: Don't know much about Persia, but that's apparently from Greek sources probably recorded by the Seleucids.

The Holy Roman Empire: That's 1789. The same year the French Revolution happened. There are more than enough sources about the HRE, which was very literate, in 1789.

The Aztec Empire: There's The True History of the Conquest of New Spain and The Conquistadors, all primary Spanish sources. Lienzo de Tlaxcala and Historia de Tlaxcala written by Tlaxcalan allies of the Spanish allows us to know the rough borders of Tlaxcala. There are also lengthy accounts from the Mexica viewpoint, as translated by friars, and actual texts written in Classical Nahuatl. The Nahua and Texcoco recorded events.

Regarding Persia, /u/Hinmatoowyalahqit is incorrect in highlighting the Seleucid Empire. We simply do not have primary sources from the Seleucid period regarding the extant and territories of the Persian Empire.

/u/Spoonfeedme, on the other hand, has the right idea. Herodotus does indeed contain the greatest amount of information regarding the extent of the Persian Empire and its various satrapies. Nonetheless, we also have a royal Persian incription, the Behistun Inscription. Commissioned by Darius the Great in the late 6th century BCE, this inscription and relief portray the numerous peoples and land that Darius ruled over, and the various tribes contained therein. Many other inscriptions of Darius corroborate this data. The exact boundaries are not known, but, as /u/Spoonfeedme explained, we can get a decent sense from parallel archaeological and textual evidence.

For the ancient Mediterranean world in general, historians use a combination of the data gleaned from ancient geographers and ethnographers (e.g. Herodotus, Pausanias, Strabo, et al.), and material evidence. Both the Greeks and Romans used their own variation of boundary stones in varying degrees at different points in history. For instance, here is an article by ancient historian Josiah Ober explaining the uses of Greek horoi as boundary delimiters.