How is the historical accuracy of the Bible defended?

I'd like to start out by pointing out that history is a murky subject. We're all familiar with the saying that "history is always written by the winners". There are plenty of things that are accepted as historical fact that are either in dispute, or that are later proved false by new evidence. Just because something is regarded as reliable, accepted historical "fact" does not mean that it is, indeed a "fact" or Truth. This applies equally to secular history and the historical accounts recorded in Scripture.

There is another issue here, as well, in dealing with supernatural events as described in Scripture. It's covered pretty well at http://depts.drew.edu/jhc/hartlich.html in the "Thesis 2" section. In summary, it tells us that, as applied to Sacred Texts there is a limit to how much we can actually verify. It gives the example of Matthew 28 as follows:

Here it is related that — as the women came to the grave — a great earthquake took place, "for an angel of the Lord descended from heaven and came and rolled back the stone, and sat upon it. His appearance was like lightning, and his raiment white as snow. And for fear of him the guards trembled and became like dead men. But the angel said to the women..."

Several paragraphs of detail go into the fact that we could, theoretically, verify that an earthquake took place, but not that it was caused by an Angel of the Lord. We could also not verify the appearance of said angel, nor the trembling of the guards. (But neither could we refute it.)

So, from the outset, the apologist acknowledges that we cannot prove or disprove the supernatural nature of events. We are limited to those things, and those things only that can be proved historically.

Failure to recognize this is to set us up immediately for straw-man attacks, and to be accused of circular reasoning.

That said, the first rule in determining historical accuracy to those events that we can verify is to use the same rules that would be applied to secular historical documents. Using a different set of rules to determine historical accuracy of Scripture is dishonest, and shows a bias that is not acceptable in applied historical methodology.

As much as I hate using Wikipedia as a source, the core principles are defined here in layman's terms much better than what I can find at other sources.

Core principles

The following core principles of source criticism were formulated by two Scandinavian historians, Olden-Jørgensen (1998) and Thurén (1997):1

• Human sources may be relics such as a fingerprint; or narratives such as a statement or a letter. Relics are more credible sources than narratives.
• Any given source may be forged or corrupted. Strong indications of the originality of the source increase its reliability.
• The closer a source is to the event which it purports to describe, the more one can trust it to give an accurate historical description of what actually happened.
• A primary source is more reliable than a secondary source which is more reliable than a tertiary source, and so on.
• If a number of independent sources contain the same message, the credibility of the message is strongly increased.
• The tendency of a source is its motivation for providing some kind of bias. Tendencies should be minimized or supplemented with opposite motivations.
• If it can be demonstrated that the witness or source has no direct interest in creating bias then the credibility of the message is increased.

Procedures

Bernheim (1889) and Langlois & Seignobos (1898) proposed a seven-step procedure for source criticism in history:2

• If the sources all agree about an event, historians can consider the event proved.
• However, majority does not rule; even if most sources relate events in one way, that version will not prevail unless it passes the test of critical textual analysis.
• The source whose account can be confirmed by reference to outside authorities in some of its parts can be trusted in its entirety if it is impossible similarly to confirm the entire text.
• When two sources disagree on a particular point, the historian will prefer the source with most "authority"—that is the source created by the expert or by the eyewitness.
• Eyewitnesses are, in general, to be preferred especially in circumstances where the ordinary observer could have accurately reported what transpired and, more specifically, when they deal with facts known by most contemporaries.
• If two independently created sources agree on a matter, the reliability of each is measurably enhanced.
• When two sources disagree and there is no other means of evaluation, then historians take the source which seems to accord best with common sense.

Using this framework, apologists point to the fact that the Gospels, and many of the Scriptures were written by eyewitnesses to the accounts, as well as to internal textual criticism, additional supporting information from extra-Biblical texts (where they can be found), and from archaeology.

Based on archaeology alone, we have overwhelming evidence for the historical accuracy of much of the Bible. But, as the atheist point out, this still proves nothing about the supernatural aspects of Scripture.

Fortunately, the field of apologetics doesn't deal with proving the Truth of Scripture. As stated in a different answer, Christianity assumes a measure of faith. Apologetics deals with defending what is defensible from detractors, not proving anything.

When it comes to the supernatural, apologetics doesn't say "We can prove this happened as recorded in Scripture". However, arguments can be made, and questions can be asked about which proposed version of events is the most plausible, or reasonable.

Unfortunately, that's a grey area, and personal bias and preconceived notions will likely dictate what any one individual will think is "the most plausible explanation."

Therefore, the best we can do in trying to defend the supernatural aspects of the Bible is to say "this is one possible explanation. I believe it makes sense because.... Take it or leave it." This falls outside the realm of defensible apologetics and into debates over personal opinion based on interpretation of provable facts. Those provable facts may well be within the realm of "on-topic" for apologetics, but the unprovable portions are iffy. They may offer reasonable explanations, but nothing that will satisfy your typical "I won't believe it unless you can prove it to me" skeptic.

This is not to say that providing views on why you believe the supernatural aspects is wrong, or that it doesn't have a place in witnessing, or defending your beliefs. Just know the difference between what can be conclusively proven and what needs to be accepted by faith. You can be sure that most of the atheists and non-believers do, and are ready for this type of mistake.