The "known world" was definitely a concept used by ancient geographers, historians, philosophers and statesmen. The Greeks had a specific word for this concept. They used oikoumene (οἰκουμένη) to refer to the known or inhabited world. This word is related to the verb oikeo (οἰκέω) which means to inhabit, and the noun oikos (οίκος) which means house. This concept and term was later adopted by the Romans.

What the oikemene precisely was, how large it was and how much of it was known of course depended on you asked. Furthermore, the timeframe you proposed (500 BCE to 500 CE) is of course a very long time. This was also a time in which both more empirical geographical knowledge and theoretical knowledge of the world expanded greatly. Imagine someone asking what the geographical knowledge was between the year 1000 and 2000. A lot happened in those years and geographical knowledge changed immensely.

The first thing to note is that we are hardly aware of what an 'average' Greek or Roman citizen thought about the known world. Most people were then, as is the case now, not geographers or philosophers and it is not hard to imagine that most people weren't too concerned with the specifics of how big the world exactly was. If you would ask a European or American citizen today how big the universe was, they would probably answer: "very big" and have some idea of solar systems and galaxies. An astronomer would of course be able to give you a much more exact and elaborate answer. And remember that even today there are people who believe the earth is flat!

In the ancient world it was no different. We have the ideas of people like Eratosthenes or Ptolemy about the known world, but those were the experts at the time. Their contemporary countrymen who were farmers or craftsmen in all likelihood did not hold the same sophisticated and often theoretical ideas. Outside the scope of the immediate world they encountered they may have had more mythological notions of distant lands, or some might not even have cared to think about that stuff at all. Unfortunately we can't say anything for sure, since the only sources we have are of upperclass or intellectual writers.

Secondly, what was known about the world did not only depend on when or who you asked, but also where you asked. Today, scientists work together and share ideas on a global scale. The ancient world was a lot more fragmented, especially before the Roman domination of the Mediterranean world. Carthaginian sailors had lots of information about the Atlantic coasts of Europe and Africa, Greeks merchants knew about the lands surrounding the Black Sea and the Persians controlled the area between the Hindu Kush and the eastern coast of the Mediterranean. But they did not work together and mostly were not, or only vaguely, aware of what those other people knew.

The Roman conquests did a lot to consolidate this knowledge from different sources. But still. According to the Romans the Mediterranean was obviously the centre of the world and the Persian world lay at their eastern periphery. Of course, the Persian kings equally thought they ruled the heartlands of civilisation with the Romans occupying a marginal area to their west. Indian kingdoms most likely also thought they were at the centre. And of course the Chinese emperors at the time knew for sure that they ruled "all under heaven" from the middle of the world and all other areas were peripheral. A still common Chinese way to refer to China is Zhongguo, which can be translated as land or kingdom of the middle.

Now, going back to our familiar Mediterranean Sea, how large did the Greeks and Romans actually think the world was? One of the earliest ideas was that the inhabited world was a landmass on a flat disc surrounded by an immense river called Okeanos. Starting from the 7th and 6th centuries BCE, Greek philosophers however created more and more elaborate theories about the world.

Pythagoras and Parmenides believed the earth was a sphere and the oikoumene lay at the perfect location between uninhabitable hot and cold climate zones. In Plato's dialogue Phaedo we can read Socrates saying that: "[...] the earth is round and in the middle of the heavens [...] I believe that the earth is very large and that we who dwell between the pillars of Hercules and the river Phasis [meaning the lands around the Mediterranean Sea] live in a small part of it about the sea, like ants or frogs about a pond, and that many other people live in many other such regions." So in the fourth or third century BCE at least some people already had the idea that the world was a large sphere and the known world was just a small part of it.

Geographical knowledge expanded first by Alexander's conquests, and later by the Romans. Alexander's expedition meant the east was "opened up" to the Greek world. Greeks traded and settled in lands as far as modern day Afghanistan. Later the Romans did the same for the west with the conquest of Carthaginian North-Africa and Iberia and Celtic Western Europe, all the way to the British isles.

Under the Romans all these different geographical traditions merged. The geographers Eratosthenes, Strabo and Ptolemy ultimately consolidated all this knowledge. Ptolemy's texts continued to be the top geographical texts well into medieval world, even up until the age of exploration when Africa was circumnavigated and the Americas discovered.

Finally, I'll try to give a direct answer to your question. The Greeks and Romans (well, at least some of them) knew the earth was a sphere. Several philosophers and geographers tried to calculate its size, with varying accuracy. The most successful was Eratosthenes who calculated the circumference of the earth by measuring the shadow of a stick at noon both in the south of Egypt and in its northern capital of Alexandria. He came very close, possibly only with a margin of error of less than a thousand kilometres.

Since they knew all this, it must have been obvious that the inhabited world was only a small part of the whole world, as the quote of Socrates illustrates. Knowledge of distant eastern lands certainly existed, although somewhat vaguely. Speculation about lands west of the Atlantic Ocean or inhabitable places past the Sahara desert on the southern hemisphere remained exactly that, speculation. They knew it was possible, since the earth was big enough, but never were able to explore beyond Eurasia and northern Africa.

So, at the height of the Roman Empire, the known world was several thousand kilometres from west to east, with the eastern world only vaguely known and accurate distances past the Persian empire not known. The northern border of the empire were the dense forests of Germania and Dacia and the steppes of Eastern Europe and Central Asia. The southern border was of course the vast Sahara desert. The distance between Germania and the Sahara is probably something like two thousand kilometres, depending on where you start measuring.

All in all, Socrates had it right when he said the world was very large and that people lived around the Mediterranean Sea as frogs around a pond, most of time quite oblivious to the larger world. This remained true for nearly two thousand years.

Edit: For anyone that is interested, a very interesting and equally accessible book about this very subject is D.W. Roller's Ancient Geography. The Discovery of the World in Classical Greece and Rome. Another interesting (and free!) source is the book series The History of Cartography. Volume 1 focuses on the Mediterranean world and can be accessed online for free here.