What about the "engineering impossibilty for the times" of the Egyptian pyramids?  Without modern machinery, there is no way to explain their construction.

In Pompei, the city virtually destroyed by the eruption of Mt. Vesuvius, there were crossing bars on streets that mechanically came down, wealthy people's homes were constructed with a type of air conditioning, the baths were kept hot by means of cleverly constructed lower areas where fires burned--and somehow there was enough oxygen for them!

Road construction was extremely advanced in Rome, aqueducts and other advanced forms of bridges were not seen in Europe for centuries.

Also, the ancient Greeks were so mathematically advanced with Pythagoras who gave the world the Pythagorean theorem which is yet employed today.  There is evidence that the Babylonians understood this therem, as well.

The precision with which some of the ancient structures, in particular the pyramids, were executed is amazing, but I think the ability to organize and use the amount of labor it must have taken to build the pyramids and other structures is really astonishing. I believe the scholarly view now is that the builders of the pyramids were not really slaves, but volunteers and conscripts. That an ancient state was developed enough to control such a project over such a long period of time is amazing. 

The Romans were also the first people to use concrete.  But really, their ability to create the sorts of structures that they did without calculators, let alone computers, was the most amazing thing about them.  Also, the sheer determination of people was impressive.  The amount of effort, for example, that went into making the pyramids is amazing.

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.