I know the oldest books from the Bible that we have are basically scraps. That means that what we are reading was recorded more recently than the dating of those pages. So what is the oldest complete version of the Bible that exists? What is the earliest date we could get if we built a Bible from each of those aforementioned papers? (I know this question probably can't be answered because that would mean someone figured out what the earliest date of each pieced together book of the Bible was)
It depends what you mean by 'complete', but probably Codex Sinaiticus would make a good contender. Mid 4th century, a complete text of the New Testament (minus some verses here and there, but 'complete' as a document; that's just the nature of all ancient biblical manuscripts).
The Old Testament portion is incomplete, and it's the Septuagint. So it includes some books considered apocryphal by Protestants, deutereocanonical by Roman Catholics. In particular the Pentateuch has not survived well.
The oldest complete Hebrew text of the Old Testament is the Leningrad Codex, dated to 1008. The Aleppo codex would be older (early 10th century) but has been incomplete since it disappeared in 1947 and then reappeared but missing portions in 1958.