It is true that we do not have the original texts of the books of the Bible. Nevertheless, we can reconstruct the original texts because we have thousands of copies of the originals that we can examine and compare. There are far more early manuscript copies of the biblical texts than any other work of ancient literature. With the discovery of the Dead Sea Scrolls, we now have copies of the Old Testament that go back to the time of Jesus. Before this discovery, our earliest copies of the Old Testament dated to around 900 AD. The Dead Sea Scrolls, however, proved that the text had not been substantially altered in those 900 or so years.
With regard to the New Testament, we have around 5,700 Greek manuscripts containing either the entirety or fragments of the New Testament. Some fragments of the New Testament date to within 50 years of the original text, copies of entire New Testament books date to within 100 years of the original, and we have nearly complete copies of the entire New Testament that date to within 150 years after it was written. For comparison, consider that Homer’s Iliad has the second most number of copies at 643 (far less than 5,700), and the earliest of these copies is not dated until 400 years after the Iliad was written. For more perspective, consider that we only have seven copies of the works of Plato, and the earliest is 1,300 years after the original was written. (A number of texts and websites provide this information)
The copies we have of the New Testament are far earlier and far more numerous than that of any other ancient text. Because of this, scholars are able to reconstruct the original text with great accuracy by comparing all of the manuscripts. Although there are variations between the copies, the vast majority of these variations are mere differences in spelling. When the copies differ in terms of the actual words that are used, we have enough manuscripts to compare so that, in almost every case, it is possible to determine what the original document said with no doubt whatsoever. Just a fraction of a percent of the text is in dispute, and none of these disputed texts affects Christian doctrine.