Not very, though even so people often massively over estimate the similarities. I'll try to summarise everything for you.

In classical Greek you had several dialects of Greek of varying inter-relatedness and mutual intelligibility. So when we say "ancient Greek", even if you ignore the temporal problems (the Greek of Homer, Classical Athens and the Bible are all "ancient" to us but differ from each other in varying ways) we're still basically imposing a modern category on a reasonably complex situation.

Languages, all languages, change over time and Greek has been subject to quite a bit of lexical, phonological and morphological change. What makes matters more complex has been consistent attempts at either Atticising (e.g importing features from Attic Greek) or just all around Classicising (generally being more classical) at varying points in history. The Bible too has had a large effect. This distorts the picture.

A few people are going to comment on iotacism, whereby several sounds in ancient Greek gradually merged together into the /i/ sound. Here's a fun example:

οι υιοι εχοιεν την υγιειαν

In ancient Greek you'd get something like: hoi huioi ekhoien ten hugiean and in modern: i ii ekhien tin igiian.

There have been other just as interesting widespread changes at various points though, the shift from a pitch to a stress based accent, the loss of vowel quantity, consonants shifting to become harder, a loss of a case, the reorganisation of the verbal system, reduction of moves, loss of participles in any real sense, particles of speech and so on and forth.

So, no, short answer of course they're not mutually intelligible. If you want to know more see if you can get hold of Horrock's "A History of the Greek Language and its speakers", also check out the wikipedia article on the Greek language question.

Wow I half suspect I've typed a gigantic messy wall but I'm shattered and am going to drop so good night, sorry I'm not more helpful.