There are people who don't believe that God can become a man, as we Christians say He did as Jesus. Is there any evidence that God is able to put on flesh in the Old Testament? What logical arguments for the possibility can we offer?
In the Old Testament God does take physical forms however the incarnation of the Messiah is unique. Hypostatic union means human and divine natures united in one person. It would be misleading to say the OT appearances of God as an angel, man or tree are hypostatic union. Some helpful references from the tanakh - The Hebrew Bible Teaches Complex Monotheism Long Before Greek Philosophical Religion :
In the following quotes, Sommer points to God’s visit with Abraham in Genesis 18-19 as an example of early biblical teaching attesting to the one, true God’s multiplicity of personhood. Genesis 18, a J text, provides one of the most revealing cases. At the outset of that chapter, we read, “Yhwh manifested Himself to Abraham amidst the trees of Mamre while Abraham was sitting at the entrance of his tent, at the heat of the day. He lifted up his eyes and saw three men coming toward him” (Genesis 18.1-2). The juxtaposition of these two sentences (which are from a single Pentateuchal source) implies that Yhwh appears in the form of three men, or, at least in the form of one of the three men. 8 Abraham, however, does not realize that his visitors are not human. He directs his attention especially to one of these men, whom he addresses in the singular, using the obsequious courtesy normal in the ancient Near East: “My Lord, if you find me acceptable, please do not pass by your servant” (18.3). All three men subsequently speak in 18.9; in 18.10 one visitor, still not identified explicitly, predicts or promises to return months later, at which time Abraham will have a son. Thus this visitor speaks prophetically, which is to say, in God’s voice, though whether this is because the visitor is God or merely represents God is not made clear. (The alternation between singular and plural continues throughout this passage.) Finally, in 18.13 the narrator stops being coy and simply refers to one of the visitors as Yhwh. Two of the visitors leave, and the one who remains with Abraham is now clearly identified as Yhwh (18.22); Abraham’s knowledge is now parallel to the reader’s, for in the discussion that follows it is clear that Abraham now knows who the remaining Visitor is. The other two beings are subsequently refer to as angels (19.1) It is clear that Yhwh appears in bodily form to Abraham in this passage; what is less clear is whether all three bodies were Yhwh’s throughout, or whether all three were Yhwh’s at the outset of the chapter but only one of them is by its end, or whether the other two were merely servants (perhaps human, perhaps divine) who, for no clear reason, were accompanying Yhwh. In any event, the being who certainly was Yhwh was less than the deity’s full manifestation. The visitor was not recognizable as God to Abraham at the outset, and he (He?) acts with a humility unbecoming a deity as h/He stands waiting before Abraham (at least according to what even traditionalist scholars regard as the original text of verse 22). Further, even though the visitor is clearly identified as Yhwh by the middle of the chapter and refers to God in the first person while speaking, h/He announces h/His intention to “come down” from heaven to observe Sodom and Gomorrah in verse 21 – even though H/he is already down on earth at this point. This visitor clearly is and is not identical with Yhwh; more precisely, He is Yhwh, but is not all of Yhwh or the only manifestation of Yhwh; rather, He is an avatar, a “descent” of the heavenly God who does not encompass all of that God’s substance. – Benjamin D. Sommer, The Bodies of God and the World of Ancient Israel, p. 40-41
A similar phenomenon occurs in the famous J narrative in Genesis 32 in which Jacob wrestles with a being initially described simply as a “man” (32.25). One soon senses that this man is in fact some sort of otherworldly being, because he cannot remain on earth once the sun rises and because his name is a secret. (It is perfectly normal to find a text referring to an angel as an [man] in the Hebrew Bible; see Genesis 18.2, 19.1, Judges 13.16; Zechariah 1.8, 11; Daniel 9.21.) Jacob names the place of this encounter Peniel (“face of God”), saying “I have seen ‘elohim face to face, yet my life was saved” (32.31). The word 'elohim' can refer both to a lower ranking divine being (or angel) and to the God also known as Yhwh, and it is not clear which meaning the text intends here. Hosea 12.4-6, interestingly takes it to mean both as it summarizes this story in poetic parallelism: “In his might he wrestled ‘elohim, He wrestled an angel (mal’akh) and prevailed….It was Yhwh, the God of hosts; Yhwh was His name.” One might initially suggest that in the first of these lines the word 'elohim' means the plural noun “divine beings” and not the singular noun “God,” but the text goes on immediately to identify the 'elohim:' “It was Yhwh…” (12.6). In other words, in Hosea 12 the being who wrestled with Jacob was not a mal’akh who also could be called an 'elohim;' rather, it was the God Yhwh, who can also be termed a mal’akh. The reason for the apparent confusion between God and angel in these verses from Hosea is simply that both passages, Hosea 12 and Genesis 32, reflect a belief that the selves of an angel and the God Yhwh could overlap or that a small-scale manifestation or fragment of Yhwh can be termed a mal’akh. – Benjamin D. Sommer, The Bodies of God and the World of Ancient Israel, p. 41
Evidence of this sort of angel is not limited to the Pentateuch. In the story of the commissioning of Gideon, a mal’akh appears to Gideon from underneath a tree and speaks to him (**Judges 6.11-13), but as the story progresses, we are told simply “Yhwh turned to face him and said…” (verse 14).** Like Abraham in Genesis 18, Gideon at first does not realize that his visitor is divine. As James Kugel observes Gideon “is certainly unaware that this is “the angel of the LORD,” or else he would do what everyone else does in such circumstances, bow down in reverential awe. Instead, he fixes on the stranger’s pious greeting in order to give him a somewhat impious retort [which can be paraphrased:] “Oh yeah? If the LORD is with us, where is He now?” Then the angel turns to him and says, “Go in this strength of yours and save Israel yourself from the Midianites – am I not the one who is sending you?” Certainly this should have tipped him off: who could this “I” be if not God Himself? Yet it is only after the next exchange, when he is told, “But I will be with you and you will defeat the Midianites to a man,” that Gideon begins to suspect that the visitor is not an ordinary human. Even so, he is still not sure: he wants proof, a sign…It is only after the flame magically consumes the offering and the angel himself disappears that Gideon’s moment of confusion may truly be said to be over.” The reader may share some of Gideon’s confusion. The text variously identifies the speaker as Yhwh (verses 14, 16) and Yhwh’s mal’akh (12, 20, 21). Indeed, Gideon’s visitor sometimes speaks in the first person of God (verses 14, 16) and sometimes in the third (verse 12). One might want to argue that Yhwh was located in heaven and spoke through a lower ranking divine being sent to earth with a message, but it is specifically Yhwh who turns His face toward Gideon in verse 14. At the same time, we are told (verse 22) that Gideon saw Yhwh’s mal’akh, and even though it was Yhwh who turned to face Gideon, it was the mal’akh who left the place (verse 21). The text seems self-contradictory only if one insists that an angel is a being separate from Yhwh. On the other hand, if one can understand angel as a small-scale manifestation of God or even a being with whom Yhwh’s self overlaps, the text coheres perfectly well. – Benjamin D. Sommer, The Bodies of God and the World of Ancient Israel, p. 42-43
The Old Testament Christophanies (like the man appearing to Joshua in Joshua 5 as the "Commander of the army of the Lord") are clues that God might do this (become a man), but these should be combined with prophecies that prove that God MUST do this. Throughout the Bible, the prophets add one feature after another to the description of the coming Messiah. As time went on, the picture began to come into focus. At first, it appears that these are merely pointing to a special man, and giving clues as to when he will appear (such as Daniel's prophecies). But certainly, by the time you get to Isaiah, this messiah is described as accomplishing things that only God can do.
"But He was wounded for our transgressions, He was bruised for our iniquities; The chastisement for our peace was upon Him, And by His stripes (whip lashes) we are healed. And the LORD has laid on Him the iniquity (sin) of us all." (Isaiah 53:5-7)
The many prophecies unequivocally show the messiah to be of human descent, proving his humanity. The ability to save people from sin, and to rise from the dead show his necessary divinity:
Therefore my heart is glad, and my glory rejoices; My flesh also will rest in hope. 10 For You will not leave my soul in Sheol, Nor will You allow Your Holy One to see corruption. 11 You will show me the path of life; In Your presence is fullness of joy; At Your right hand are pleasures forevermore. (Psalm 16)
And from Proverbs 30, we see a glimpse of his ability to travel between Heaven and earth at will:
Who has gone up to heaven and come down? Whose hands have gathered up the wind? Who has wrapped up the waters in a cloak? Who has established all the ends of the earth? What is his name, and what is the name of his son? Surely you know!