There are many reasons why the ancient world is amazing. Let me name a few of them.
First, if you look at the Romans, you will realize that they were excellent engineers. In fact, many of their structures still exist and are still in use. For example, people still use Roman roads and bridges. They built things to last. When we include ancient structures, like the pyramids, there is even more amazement.
Second, if you look at ancient literature, we still read them with great profit. This is another incredible fact. Think of Plato, Socrates, Aristotle, Confucius and the like. Some would say that their intellectual powers have not been surpassed.
Third, the ancients were also great scientists. They created things like geometry and made incredible calculations without computers.
Fourth, the ancient accomplished these great feat without modern technology. Hence, they showed great innovative powers.