Learn Turkish: On altı adam yaşlı değil.

In Turkish, numbers from 11 to 19 are formed by simply combining the word for ten (on) with the single digit. So, on (ten) plus altı (six) becomes on altı (sixteen). It is much simpler than English!

Since there are sixteen men, why don't we use the plural word adamlar?

This is a very important rule in Turkish: when you put a number before a noun, the noun always stays in its singular form. Because on altı already tells us there is more than one, adding the plural ending -lar to adam is strictly incorrect.

I have seen the word eski used for "old". Could I use it here instead of yaşlı?

No, because Turkish has two different words for "old". You use eski for objects and things (like an old book), but you must use yaşlı for people and animals. Yaşlı comes from the word yaş, which means "age".

Where is the verb "are" in this sentence?

Turkish doesn't have a standalone verb for "to be" (am/is/are). Instead, it uses endings on the last word. For the third person plural ("they"), the ending is often left off entirely. Saying değil is perfectly correct here, though you might sometimes see değiller.