Of course, it is impossible to demonstrate the inerrancy, or even the inspiration, of the Gospels using the methods of history. However, we can present a strong case for the reliability of the Gospels. Even if he cannot bring himself to treat the Gospels as inspired, the sceptic should acknowledge that the Gospels contain good, historical information about the life of Jesus.
NT Wright explains why New Testament scholars are now much more likely to treat the Gospels as good, useful sources for the life of Jesus:
For those seeking to go a little deeper, Richard Bauckham explains why the Gospels were not produced by a process analogous to “Chinese Whispers”
Bauckham has an excellent essay on his research here.
Ben Witherington III explains how the Gospel writers could obtain reliable information about Jesus. ( The Gospels were written within living memory of Jesus, in a culture which prized memorization and oral tradition.) Greg Boyd explains how many details in the Gospels support their historicity. For those with extra time, Craig Keener has an excellent essay explaining why we should treat the Gospels as reliable documents.
The Book of Acts also emerges as reliable in Keener’s opinion; this is significant as Luke and Acts had the same author.
Alan Millard points out that writing was more common than we might suppose. Indeed, we need to understand that oral traditions generated written notes as aids and guides, as Greg Boyd explains. Christopher Tuckett points out that the Gospel writers did not simply depend on oral sources, but written sources also. It seems likely that Luke and Matthew quite carefully copied information from Mark (and at least one other source); so the Gospel authors did not treat their sources carelessly.
Finally, Philosopher Tim McGrew has provided a robust and interesting case for the reliability of the Gospels, drawing on modern research and on the writings of older Christian apologists. These lectures repay careful listening.