I don't speak Greek, so I don't know if a contemporary Greek person is able to read Aristoteles and to note it is in Greek, apart from the issue that to read a philosophical text is not an easy task, even in one's native contemporary language. When I read in the bible it is very clear to me that it is in Hebrew, but I have read in the bible since I was little child, and I'm quite used to it. The grammar is different, the vocabulary is different, and even native Hebrew speakers need to translate some word, and sometimes hardly "decrypt" it; especially in its literaturic parts (eg Psalms). However, note that regarding the bible, one only needs lecture abilities, and he is not expected to talk/listen/write in it.

The language of the Bible is more like an archaic form of the language much like you find in the translations available in English. It is full of very flowery language, a bit like Shakespeare. So you can learn to understand it, much as school kids learn to understand Shakespearean English. As Geri says you only have to recognize and understand passively. Modern Hebrew (including slang) is based on this ancient form of the language and shares many expressions, but modern everyday language is more straightforward and easier to learn to speak.Of course if you also learn Biblical Hebrew your modern Hebrew will only be the richer for it.

The good news is, the basics are the same. The alphabet is the same, the morphology is mostly the same (except for some exceptions), the genders, prepositions and pronouns are all the same. So if you start on one, you'll be essentially starting on the other at the same time. The bad news is that there are some big grammatical differences. Nowadays, the avergae Hebrew-native child who hasn't been exposed to the Torah since early childhood and is not well-read simply cannot understand what is being said - and that's the best-case scenario. The worse-case scenario is that they think they can understand it, but actually understand nothing. Sentence order is different; tenses work differently; vocabulary has changed meaning. Add to that that the grammar throughout the Torah is different: the grammar in Genesis is subtly different from the one used in Deuteronomy (and it only gets worse in the rest of the Tanakh). You'd need someone to point out where the differences are, what they are, and what their Modern Hebrew equivalent is. The equation of Modern-Biblical Hebrew to Modern-Early Modern English is pretty spot on, except Hebrew is a language on steroids and develops new grammar as fast as you can shake a stick at it. Personally I think Modern Hebrew would be simpler for someone from an English base to learn, because all the European immigrants have affected it some in the direction of Indo-European languages, and because you'd have better luck snagging native speakers to ask questions. (Then again, I'm biased.)