What is the biblical basis for the historicity of the Bible?

That professor hasn't read the Bible, apparently.

Eyewitness Account

Luke 1:1-4 (my translation)

Inasmuch as many have undertaken to compose a narrative of the things which have taken place among us, (even as it was handed on to us by those who were eyewitnesses from the beginning, and those who tended to [the matter of preserving] an account), it seemed good to me also, having followed all things closely from the beginning, to write to you an orderly account, most excellent Theophilus, that you might know the verity of the things that have been told you.

Luke even uses the adjective ακριβως (accurately; closely; scrupulously). Luke claims his Gospel is an accurate history of Jesus.

Real Old Testament People

As for the Old Testament, it is confirmed by its affirmation in the New: by Jesus, and by the geneology listed by Luke.

Luke 3:23-38 (DRB)

And Jesus himself was beginning about the age of thirty years; being (as it was supposed) the son of Joseph, who was of Heli, who was of Mathat, 24 Who was of Levi, who was of Melchi, who was of Janne, who was of Joseph, 25 Who was of Mathathias, who was of Amos, who was of Nahum, who was of Hesli, who was of Nagge, 26 Who was of Mahath, who was of Mathathias, who was of Semei, who was of Joseph, who was of Juda, 27 Who was of Joanna, who was of Reza, who was of Zorobabel, who was of Salathiel, who was of Neri, 28 Who was of Melchi, who was of Addi, who was of Cosan, who was of Helmadan, who was of Her, 29 Who was of Jesus, who was of Eliezer, who was of Jorim, who was of Mathat, who was of Levi, 30 Who was of Simeon, who was of Judas, who was of Joseph, who was of Jona, who was of Eliakim, 31 Who was of Melea, who was of Menna, who was of Mathatha, who was of Nathan, who was of David, 32 Who was of Jesse, who was of Obed, who was of Booz, who was of Salmon, who was of Naasson, 33 Who was of Aminadab, who was of Aram, who was of Esron, who was of Phares, who was of Judas, 34 Who was of Jacob, who was of Isaac, who was of Abraham, who was of Thare, who was of Nachor, 35 Who was of Sarug, who was of Ragau, who was of Phaleg, who was of Heber, who was of Sale, 36 Who was of Cainan, who was of Arphaxad, who was of Sem, who was of Noe, who was of Lamech, 37 Who was of Mathusale, who was of Henoch, who was of Jared, who was of Malaleel, who was of Cainan, 38 Who was of Henos, who was of Seth, who was of Adam, who was of God.

This is airtight. A genealogy back to the first man, beginning with Joseph. Ignoring the 'whose genealogy is it' question, it claims Joseph is literally a descendant of Adam, the first man created by God. Now unless Joseph is a literary device—an untenable joke—then the Old Testament descriptions of the patriarchs is literally true.

Likewise, we see in 1 Peter, for example, claims that those before the Deluge went to some spiritual prison (cf. Lk. 16:22; Heb. 11:40), where Jesus went to preach the Good News after He yielded up His spirit on the cross (Mt. 25:50).

1 Peter 3:18-20 (DRB) Because Christ also died once for our sins, the just for the unjust: that he might offer us to God, being put to death indeed in the flesh, but enlivened in the spirit, 19 In which also coming he preached to those spirits that were in prison: 20 Which had been some time incredulous, when they waited for the patience of God in the days of Noe, when the ark was a building: wherein a few, that is, eight souls, were saved by water.

Cf. 2 Pet. 1:16.

Real people at the time of Noah, when the ark was being built.

Even Unto This Day

There is a recurring phrase in Old and New Testament: "even to this day." It means that whatever happened in some former time remains true "even to this day." A New Testament example is Matthew 28:8:

Matthew 28:8 (DRB) For this cause the field was called Haceldama, that is, The field of blood, even to this day.

There is a direct causal relationship between the event described (and fixed by its context at some explicit time in history) and the naming of a thing or person or place.

Interesting Old Testament examples, however, might include Genesis 22:13-14:

Genesis 22:13-14 (DRB) Abraham lifted up his eyes, and saw behind his back a ram amongst the briers sticking fast by the horns, which he took and offered for a holocaust instead of his son. 14 And he called the name of that place, The Lord seeth. Whereupon even to this day it is said: In the mountain the Lord will see.

The author of Genesis, at least, is convinced that this place is treated this way because Abraham himself named it after a real experience he had. He's not ignorant that he has placed Abraham in a certain era.

Cf. Josh. 7:25-26.

Other instances of this abound. E.g.

Genesis 35:19-20 (DRB) So Rachel died, and was buried in the highway that leadeth to Ephrata, that is Bethlehem. 20 And Jacob erected a pillar over her sepulchre: this is the pillar of Rachel's monument, to this day.

Or,

2 Samuel 6:8 (DRB) And David was grieved because the Lord had struck Oza, and the name of that place was called: The striking of Oza, to this day.

Cf. Vulg. Judith 16:31.

These are just a few pieces of evidence that the Old Testament was considered by both the author and first century Jews (themselves partially an evidence) to be the actual history of the Jews.