Properly understood, the doctrine of inerrancy--the teaching that the Bible contains no errors--applies only to the original copies of the biblical documents. The original writings came directly from God through human authors.
The Old Testament was originally written in Hebrew and Aramaic (cf. Genesis 31:46; Ezra 4:8-6:18; 7:12-26; Jeremiah 10:11; Daniel 2:4-7:28), and the New Testament was penned in Greek. After the Old and New Testament books were written, scribes produced and distributed copies of the original manuscripts. They took painstaking care to craft those copies by hand, long before the days of the printing press.
In God's providence, we no longer have the original documents. They disappeared over time. What we have available now are copies of those original documents--copies produced over a number of centuries. By comparing and analyzing those copies through a process called textual criticism, we are able to determine what the original manuscripts said and where variations crept into the copies. That process has confirmed that God has accurately preserved His Word for us.
The "Chicago Statement on Biblical Inerrancy" correctly notes:
Since God has nowhere promised an inerrant transmission of Scripture, it is necessary to affirm that only the autographic text of the original documents was inspired and to maintain the need of textual criticism as a means of detecting any slips that may have crept into the text in the course of its transmission. The verdict of this science, however, is that the Hebrew and Greek text appears to be amazingly well preserved, so that we are amply justified in affirming...a singular providence of God in this matter and in declaring that the authority of Scripture is in no way jeopardized by the fact that the copies we possess are not entirely error-free...Similarly, no translation is or can be perfect, and all translations are an additional step away from the autograph. Yet the verdict of linguistic science is that English-speaking Christians, at least, are exceedingly well served in these days with a host of excellent translations and have no cause for hesitating to conclude that the true Word of God is within their reach.
While you can't handle the original documents today, you can have confidence that most English Bibles faithfully represent what was contained in those documents. God has providentially preserved His Word for subsequent generations despite the best efforts of many of His enemies to eliminate it.