It is definitely not a hight front vowel ("see"), which is the value it had in Koine and has in modern Greek.
[1]: https://blogicarian.blogspot.com/2012/10/ancient-greek-food-...
So Greek was what is called a pluricentric language and had IIRC at least five major dialects (Attic, Ionic, East Greek, West Greek etc.). Of course these changed significantly over the centuries, too.
There is obviously a whole field of research into Ancient Greek phonology and they have various points of reference to determine possible pronunciation. There are also a number of surviving texts that do discuss pronunciation of some words.
However, to me the most interesting tidbit was always that they found out how to pronounce the letter "η" (which in modern Greek is similar to "ee" like "meet") when a fragment of an ancient comedy by Aristophanes was discovered in which a guy is about to be sacrificed by a priest (I think?) and the priest tells him to behave like a sheep (as a kind of proverbial "sacrificial lamb" I suppose) and imitate sheep noises (difficult to put in English but something like "baa").
This was rendered as "βῆ βῆ" in the fragment and because Greek sheep have not changed their accent over the millennia, we were able to get some insight in to how that must have sounded. ;-)
I find that really hard to believe. Surely animal sound onomatopoeia changed over 3000 years?
Think of it similar to Latin where real pronunciation was quite different to how you hear it quoted in media today, e.g. Veni, Vidi Vici sounding more like Weni, Widi, Wiki.
What you call "real pronunciation" is the pronunciation in the Classical dialect of Latin[0].
What you hear pronounced most often is the Liturgical[1] dialect, which is certainly how I learned to sing Latin in choir in college (a secular school) and I'm pretty sure the Liturgical dialect is what's most commonly taught in Catholic schools. (I'm not Catholic and I didn't attend a Catholic school, but that's my understanding.)
Given the scientific revolution's roots in the Renaissance[2], I presume many scientists use pronunciation somewhere between Liturgical and Classical Latin.
[0] https://en.wikipedia.org/wiki/Classical_Latin
That's Liturgical vs. Classical dialects of Latin.
Also worth keeping in mind that broadcast and recordings do a lot of work to reduce language drift today, over time but especially over geography. A mountainous, watery place like Greece almost certainly would have a lot of drift in accent and informal language.
[On edit: That feeling when you swipe out a long comment, hit submit, and see a fresh neighboring long comment making the same points better.]