I’m an odd special case: although I am a self-styled world expert on machine recognition of Greek, I have not studied Classics, so my vocabulary is behind those who have studied it in school or uni. (In fact, for that reason my Aeolic is probably better than my Homeric — I had to do work at the Thesaurus Linguae Graecae on recognising Sappho, while Homer was already taken care of.) So my Ancient vocabulary starts from a more naive base from some other answerers’.
Greek has been more conservative than other languages in Europe, and part of that was the influence of Ancient Greek via the Church. (In many parts of Cyprus, for example, /θ/ has changed to /x/ throughout the language; so anthropos is pronounced akhropos. With one exception: the word for God, theos.) Universal literacy in Iceland had an even starker conservative effect. The grammar has also been conservative, though there is not as much grammar as there used to be. (No dual, no infinitive, no optative, no perfect, no future, no dative, no third declension, just two conjugations.)
Bear in mind though that there was massive reimportation of Ancient vocabulary and phrases into formal Greek (in no small part to eliminate Italian and Turkish loanwords); and that Greek has an historic orthography. So Greeks now can read more Ancient Greek than they were actually supposed to; and an 18th century Greek peasant, time travelling to Ancient Greece, would have little idea what was going on.