What event posed the biggest obstacle for Jews to accept Jesus?

Answer 1

To cite a few. As a Messiah:
Jesus did not fulfill a single one of the messianic prophecies.
Jesus did not embody the personal qualifications of the Messiah.
Biblical verses "referring" to Jesus are mistranslations.
The Torah teaches One, true, indivisible G_d. (not Father, Son and Holy spirit).

More often, as the answer above notes, the biggest obstacle for Jewish acceptance of Jesus is not what he did, but what he failed to do. However, since the question is posed exclusively as to what he did, there are a few examples: He indirectly ridiculed the laws of kashrut, He performed work on the Sabbath and, Most importantly, he indirectly claimed to be demigod or divine avatar depending on people's perception.

Functions of the Jewish Messiah that Jesus failed to fulfill (this list is not exhaustive):

• Setting up a lasting and Eternal Temple (Ezekiel 37:26-28) -- Especially important since, seeing as the Temple was extant during his life, he should have been able to "keep it going eternally" if he were the Messiah. There wouldn't even have been any need to rebuild it.
• Gather all Jews back to the Land of Israel (Isaiah 43:5-6) -- This should have been relatively easy since most Jews at that time were still in the Middle East. After his death, (for unrelated reasons) the Jews were scattered all across the Roman Empire and later the entire world. Jesus did not facilitate any migration of Jews to Israel.
• Usher in an era of world peace, and end all hatred, oppression, suffering and disease. As it says: "Nation shall not lift up sword against nation, neither shall man learn war anymore." (Isaiah 2:4) -- More wars have been fought in the name of Christianity and Islam since the time of Jesus than for any almost any other single cause; and both claim him to be the Messiah.
• Spread universal knowledge of the God of Israel, which will unite humanity as one. As it says: "God will be King over all the world -- on that day, God will be One and His Name will be One" (Zechariah 14:9) -- When Jesus died, no larger population knew about God than when he was born. Still today, after many centuries of missionizing, only half of the world professes to believe in one God.
• The Messiah must be descended on his father's side from King David (Genesis 49:10 and Isaiah 11:1) -- While the Christian scriptures may allege this, they also allege that Jesus' Earthly father played no part in making him, so to speak. Therefore, he has no patrilineal lineage to speak of (save for God himself) and cannot therefore be descended from King David.
• The Messiah will lead the Jewish people to full Torah observance. -- Since the Messiah is the resolution of the problems of this world and nothing can be added or detracted from the Law, people will naturally follow the Law.
• The dead shall rise from their graves and death shall be abolished for eternal life (Isaiah 26:19 and Daniel 12:2). -- Jesus allegedly rose a few individuals from the dead, but did not raise all of the dead nor did he abolish death for everlasting life.
  

Commandments that Jesus actively violated (all quotes from the NIV). This list is not exhaustive:

• Matthew 15:11: "That goes into someone's mouth does not defile them; but what comes out of their mouth, that is what defiles them." -- This is a blanket denial of some of the laws of ritual purity and all of the Laws of Kashrut, which tell us foods to eat and which foods not to eat. This is an abrogation of the Law.
• Matthew 19:8-9: Jesus replied, "Moses permitted you to divorce your wives because your hearts were hard. But it was not this way from the beginning. I tell you that anyone who divorces his wife, except for sexual immorality, and marries another woman, commits adultery." -- Jesus specifically rebukes the Mosaic Law and denies the permissibility of divorce which is expressly permitted according to Deuteronomy ch. 24.
• Matthew 19:29: "And everyone who has left houses or brothers or sisters or father or mother or wife or children or fields for my sake will receive a hundred times as much and will inherit eternal life." -- Jesus actively supports the break-up of families, which runs counter to an entire tradition of family unity and tribal unity throughout Israelite and Jewish history.
• Mark 1:43-44: Jesus sent him away at once with a strong warning: "See that you don't tell this to anyone. But go, show yourself to the priest and offer the sacrifices that Moses commanded for your cleansing, as a testimony to them." &John 18:20-21: "I have spoken openly to the world," Jesus replied. "I always taught in synagogues or at the temple, where all the Jews come together. I said nothing in secret. Why question me? Ask those who heard me. Surely they know what I said." -- These verses show that Jesus lies (not to mention the issue with absconding truth in parables). If he said everything openly, he would not tell people to be quiet and secretive about who he was. Lying is prohibited in Leviticus 19:11. The Law being clear and accessible, not secretive, is stated in Deuteronomy 30:10-14 and Deut. 31:9-13.
• Mark 14:24-25: "This is my blood of the covenant, which is poured out for many," he said to them. "Truly I tell you, I will not drink again from the fruit of the vine until that day when I drink it anew in the kingdom of God." -- The consumption of blood is specifically forbidden in the Torah because it is the life-source of all creatures. This is made clear in Leviticus 17:10-12.