To run through the examples given by Glubb one by one:

Neo-Assyrian - 859 - 612, 247 years
Achaemenid Persia - 538 - 330, 208 years
Macedonian - 331 - 100, 231 years
Roman Republic - 260 - 27, 233 years
Roman Empire - 27 BCE - 180 CE, 207 years
Arab Empire - 634 - 880, 246 years
Mamluk Empire - 1250 - 1517, 267 years
Ottoman Empire - 1320 - 1570, 250 years
Spanish Empire - 1500 - 1750, 250 years
Romanov Russia - 1682 - 1916, 234 years
British Empire - 1700 - 1950, 250 years.

Neo-Assyrian Empire

859 BCE marks the start of the reign of Shalmaneser III, by which point Shalmaneser's two predecessors have already made Assyria the dominant power of the region. Perhaps it would be fair to place the date sometime in the previous reign, but I understand this one. 612 BCE is the fall of the capital Nineveh to a combined campaign of Babylonians and Medes. Fair choice.

Achaemenid Persia

550 BCE is Cyrus' victory against the Medes, at which point he assumed control of the Medean empire. 538 BCE might have been chosen instead as the date of Cyrus' defeat of the Babylonians, perhaps marking that as Persia removing its last challenge to hegemony. Not sure about this choice, but if we do take the earlier one it actually moves the empire's span closer to 250 years. 330 BCE is when the Achaemenid capital fell to Alexander the Great.

Macedonian Empire

Now things get weird. 331 BCE is the battle of Gaugamela, which more or less marked Alexander's defeat of Persia. Seems odd to pick a different marker for this and the end of Persia, but it's only one year apart so whatever. The end date is a problem though. Alexander's empire shattered within a few years of his death in 323 BCE. By 100 BCE Macedonia had already been a Roman province for 46 years. I'm honestly not sure of anything that happened in 100 BCE that might mark the death of the Macedonian empire. The Seleucid empire, one of the most powerful successors, had been more or less broken by the Parthians a few decades earlier, and the other big successor in Ptolemaic Egypt still had 70 years to go before Rome annexed it. Either way, Alexander's empire broke in 323, lasting just 8 years, and if you include the Diadochi its either less than 200 or more than 300 depending on which you count.

Roman Republic

The author gives some attempt at justification for splitting the Romans in to two empires like this. I don't think they're very convincing, but let's take him at his word. 260 BCE is the battle of Mylae, the first time Rome defeated Carthage at sea. It seems to me that if you're going to mark Rome's ascendancy to empire status by when it defeated Carthage then you should pick the victory in in the second Punic war. If the first one made Rome hegemon, there wouldn't have been a second in which Hannibal tore up Italy for 15 years. Hannibal's defeat in 202 BCE seems a better marker to me. 27 BCE is the proclamation of the Roman empire under Augustus. With Hannibal's defeat as the starting point, it lasted 175 years.

Roman Empire

180 is the end of the period known as the "five good emperors". The author writes: "It is true that the empire survived nominally for more than a century after this date, but it did so in constant confusion, rebellions, civil wars and barbarian invasions." the western half of the empire would last for almost three hundred more years, and the eastern half for well over a thousand more, including reclaiming most of the western half under Justinian. Roman hegemony in Europe and north Africa would not be challenged for centuries and this date makes no sense at all.

Arab Empire

I think this is mashing up the Rashidun caliphate, Umayyad caliphate, and the Abbasid caliphate prior to the Anarchy at Samarra. This entire listing is ridiculous to call a single empire when he counts Rome as separate for the republic, the empire, the western empire, and the eastern empire.

Mamluk Empire

This one is fine, running from the mamluk overthrow of the Ayyubid sultanate to the Ottoman annexation of the Mamluk sultanate.

Ottoman Empire

Not sure why the author picked 1320 specifically, but the rise of the Ottomans isn't well-recorded and it was around this date so it's fine by me. The end date is utterly baffling though. In 1570, the Ottomans launched a war against basically every naval power in Europe, and they won it. How is that end of their power? They would, of course, survive until their defeat as a major power in the First World War for ~600 years.

Spanish Empire

Not sure why 1500 specifically was picked, but it's roughly when Spain got a foothold in the Americas so okay. Nothing of particular significance happened in 1750 either, though, and it would be another 58 years before the wars of independence from its colonies started (and Spain's defeat by Napoleon soon after). Spain's "lifetime" as an empire should be over 300 years.