The geographical setting for the Achaemenid Empire centred around (modern day) South Western Iran, and gradually increased in territory spreading over much of the Middle East, encompassing modern Iran, Turkmenistan, Uzbekistan, Afghanistan, western Pakistan, Tajikistan, Kyrgyzstan, southern Kazakhstan, Azerbaijan, Armenia, Georgia, Iraq, Kuwait, Jordan, Israel, Palestine, Syria, Cyprus, Lebanon, Turkey, Bulgaria, Egypt and north eastern Libya.
To put this into perspective, the Persian Empire covered an area around twice as large as the Roman Empire at its height, encompassing dozens of differing ethnographic peoples, languages and cultures. An area of approximately 8 million square kilometres (3.1 million square miles), about 50% larger than the whole of Europe, and almost as many different languages!
Remember that this was over two and a half thousand years ago, before industry and electricity, when messages were carried over land on foot, architecture was carved and built by hand, with no machinery to speed up the process. This only serves to highlight the enormous wealth and command of resources the Achaemenid Kings had at their disposal, and the scale of the power and ambition they had in order to commission the construction of their palaces and cities.
When studying the interactions between Ancient Greece and the Ancient Persian Empire, it is important to remember the scale and context. During this period, what we would later call Ancient Greece was rather a collection of individual city-states governing themselves, spread over the mainland and the islands of the Aegean. By contrast, the Persian, or more accurately the Achaemenid Empire was a vast territory, encompassing multitudes of ethnicities, cultures, deities and languages, all under the rule of one monarch.
The map above goes some way to putting into perspective the sheer size and spread of the Persian Empire compared to the Greek territories. Consider that the rulers of this vast dominion viewed the Greeks as little more than peripheral territory for acquisition; just one of its many borderlands which included such diverse geographical regions as Northern Africa, India, Mongolia and what we would now call southern and eastern Europe.
What is perhaps more remarkable is that the Greeks were able to hold them off as long as they did, at Thermopylae, and in fact to defeat their Persian invaders at Salamis, Plataea and ultimately, under Alexander the Great, altogether. In less than 150 years after the Persian invasion, Alexander was able to seize control of virtually the entire empire and divide it up as a new Greek empire.