Not to be confused with hebrews, the book of Hebrews in the New Testament.
Other than a few brief sections in Aramaic (Ezra 4:8-6:8; Daniel 2:4-7:28; Jeremiah 10:11; and two words in Genesis 31:47), the Hebrew Bible is written in "Biblical" or "Classical" Hebrew. The chronological period represented by these writings spans several centuries.
Although the Dead Sea Scrolls attest to a form of the text very close to that found in modern printed Bibles at the time of the turn of the era, this was still largely a consonantal text only. The apparatus of vowels and accents found in printed Bibles today only arose during the early Middle Ages (roughly 600-950 CE). The earliest and greatest examples of this work known are the Aleppo Codex and so-called Leningrad Codex.
The quantity of written Hebrew surviving from the time of the Tanakh itself is relatively small, with the best examples being the caches of letters found at Arad and Lachish, and possibly the oldest being the "Gezer calendar" from c. 10th C. BCE.
For full details on the nature of the language of this period as written in the Hebrew Bible, see the Encyclopedia Judaica article on "Hebrew - Biblical".