What do modern biblical scholars think about the Bible's reliability?

First of all, biblical scholars are not supposed to bring their personal religious beliefs to any evaluation of the Bible. Christopher Gilbert says, in A Complete Introduction to the Bible:

A person who approaches the Bible academically neither assumes that the Bible neither is divinely inspired nor assumes that the Bible is not divinely inspired, but instead remains neutral on this issue.

Gilbert goes on to evaluate the Old Testament book by book and story by story. For example, he says:

From the academic perspective, neither the biblical accounts of prehistory nor the stories about the ancestors can be regarded as historical in the strict, modern sense of that term. That is, most biblical scholars do not regard the stories contained in the Book of Genesis as presenting an objective, fact-based description of historical events.

Elliott Rabin says, in Understanding the Hebrew Bible:

There is considerable uncertainty whether the patriarchs in the Bible actually lived or are instead legends of ancestral founders. No evidence for their existence has been found independent of the Bible. Of course, we have records of very few people from ancient history, though we might have expected some independent record considering the importance the Bible attributes to the patriarchs.

Another perspective comes from Francesca Stavrakopoulou, in King Manasseh and Child Sacrifice: Biblical Distortions of Historical Realities:

The Book of Chronicles seems to be particularly problematic for modern scholarship. The problem with Chronicles is not the fact that it offers an account of Israel’s past alongside that of Kings, nor even that this story of the past can, in places, differ from and even contradict the version offered by Kings. Rather, the problem with Chronicles is that these differences and contradictions occur despite the great probability that the Chronicler has composed his history using as his weightiest sources versions of the books of Samuel and Kings. This is problematic for some scholars for it casts into doubt the historicity of the Chronicler’s story and that of his sources, which often appear to be manipulated, distorted and even disregarded by the Chronicler. But this should not be seen as a problem. Rather, the freedom with which the Chronicler appears to employ his source material emphasizes the tendentious nature of all biblical texts, reminding the reader that both Kings and Chronicles are not history writing in the modern sense, but rather are ideological stories of the past told in order to account for the present and to advise for the future…As such, Chronicles is best understood as “theocentric historiography”, or as a “theological essay”.

Francis Watson cites Philo of Alexandria, in ‘The fourfold gospel’, published in The Cambridge Companion to the Gospels (edited by Stephen C. Barton):

The Jewish theologian Philo of Alexandria (died c. 50 CE), who had argued that the spiritual or theological truth of a scriptural text need not be undermined by its literal or historical falsehood. There was, for example, no such creature as the highly intelligent talking snake of Gen 3, and yet its non-historicity does nothing to diminish the theological value of the story. Indeed, historical implausibility serves as a positive indication of theological significance.

Uta Ranke-Heinemann discusses some of the inaccuracies in Acts of the Apostles and says, in Putting Away Childish Things:

The whole book is a work of propaganda aimed at Gentile Christians and Gentiles who have not yet become Christians.