What evidence is there within the Bible that a man can or cannot see God?
We know that Genesis 32:30 unambiguously says that Jacob saw God: "And Jacob called the name of the place Peniel: for I have seen God face to face." We also know that Exodus 33:11 says that Moses saw God face to face: "And the Lord spake unto Moses face to face, as a man speaketh unto his friend."
Genesis 2-3 does not actually say that Adam and Eve saw God, but it seems hard to imagine otherwise. In 2:19, God brought each animal to Adam to be named. In 3:8, they heard God "walking in the garden in the cool of the day and hid themselves from him.
Against this, Exodus 33:20 says:
And he said, Thou canst not see my face: for there shall no man see me, and live.
In Isaiah chapter 6, Isaiah says he saw God, but most modern commentators describe this as a 'vision'. Yet, Isaiah fears he will die because he saw the Lord of hosts with his own eyes (Isaiah 6:5) If this was only a vision, and if Isaiah did not really see or speak with God and if he did not really see any seraphim, then the entire chapter seems to be a nonsense.
In Acts 7:55-56, Stephen looked up and saw God in heaven:
But he, being full of the Holy Ghost, looked up stedfastly into
heaven, and saw the glory of God, and Jesus standing on the right hand
of God, And said, Behold, I see the heavens opened, and the Son of man
standing on the right hand of God.
And in what ways can seemingly contradictory verses be explained?
Wikipedia describes the scholarly consensus that the Pentateuch (Genesis, Exodus, Leviticus, Numbers and Deuteronomy) was actually written by four major sources. Although the hypothesis continues to be debated and various improvements to the hypothesis proposed, the Wikipedia article continues to represent the essential definition of the hypothesis. Each source had a different theology and each presents us a different understanding of God. Paraphrasing the words of Gerald A. Larue, because the documentary hypothesis is the most widely accepted of all theories of Pentateuchal analysis, this answer will utilise, in principle, the conclusions reached by this method of research.
Genesis chapters 2-3 and Exodus 33:11 are parts of the passages written by an anonymous source now known as the Yahwist, who wrote of an anthropomorphic God with human characteristics.
It is the Elohist who tells us in Exodus 33:20 that no one can look at God and live, consistent with his portrayal of God as a more transcendent God who required obedience and was feared by his people. People could never look at the Elohist’s God, so he typically came in dreams or visions, but sometimes in the form of a cloud or a flame.
On the other hand, it was also the Elohist who wrote of Jacob wrestling with God and naming the place Peniel because he saw God face to face. There are elements of a more primitive tradition in this last story and it is suggested that in the earliest pre-biblical tradition, Jacob had been more than just human, hence his ability to wrestle with God all night. Otherwise, we can not really explain this passage.
Perhaps Isaiah chapter 6 is the easiest to explain. Isaiah portrays this event as his commission as a prophet to Jerusalem, and no doubt the fact that he saw God convinced many Judahites that Isaiah really was chosen. He needed more than just a vision, to convince the nation to accept him as a reliable prophet, and so feigned fear of death in order to emphasise the reality of his vision.
Acts 7:55-56 was no doubt intended to be that Stephen literally saw God, although we could perhaps say that this is a different kind of 'seeing' than to see God on earth. Uta Ranke-Heinemann says in Putting Away Childish Things, page 171, cites Hans Joachim Schoeps (Das Judenchristentum, p10) to say that Acts of the Apostles follows a clear didactic line and for this reason energetically cultivates the creation of legends and reshapes persons and events according to its own standards. John Dominic Crossan says in The Birth of Christianity, page 21, "Each of Luke and Acts of the Apostles, and one no more or less than the other, is theology rather than history." Finally, Raymond E. Brown says in An Introduction to the New Testament, page 320, that we can never verify the existence and martyrdom of Stephen. Thus, the explanation of St Stephen seeing God in heaven is that Stephen may never have existed, and if he did exist Acts is widely seen as so unreliable that this event never occurred.