1) When there was a Temple, sacrificing blood only applied to unintentional sin (Numbers 15:27, Leviticus 4:27, 5:14, 5:17).
2) If Jesus did indeed replace the sacrificial system by way of spilling his blood, any offering I find described in the Torah should have a blood sacrifice. But look, how wrong can a man be? Any animal sacrifice, whether it's an elevation offering or peace offering (that sounds ironic--a peace offering with sacrificing an animal, but I digress) has to have the blood completely drained because blood consumption is forbidden (Lev. 7:26-27, 17:10-14). In cases of theft, fraud, or lying to someone about a loan, they are required to fully compensate plus add a fifth to the principal (Lev. 5:21-26). Anyone who intentionally commits idolatry is cut off from the people Israel (Numbers 15:30-31) and cannot repent. Poor people were able to make flour offerings (Lev. 5:12-13) in cases of denying testimony (5:1), contaminating holy things (5:2-3), and false or unkept oaths (5:4). Aaron made atonement with incense (Numbers 17:11-13). Jewelry was used as a sacrifice (Numbers 31:50). Even Isaiah (6:6-7) took a live coal to himself to atone! The fact that not all sins in the sacrificial system require blood sacrifice means that blood sacrifice was not the only way to atone.
3) In all reality, whether or not the sin was intentional is a moot point. In the sacrificial system, any flesh-based sacrifice was done with animals only. Human sacrifice is abhorent in Jewish practice. The most notable story of this abhorrence is the Akeidah, or the binding of Isaac. At the end, a ram was replaced (Genesis 22:13) with Isaac. In Deuteronomy 12:30-31, HaShem states how He finds human sacrifice abhorrent, and He re-iterates this point in Jeremiah 19:4-6 and Psalm 106:37-38. If HaShem clearly states that human sacrifice is wrong, why would it all of a sudden be acceptable?
4) As Maimonides points out in Guide for the Perplexed (III, xxxii), animal sacrifice was a severly limited practice that the Israelites from which they were meant to be weaned. As we already laid out, animal sacrifice, and more specifically blood sacrifice, was limited in context. The practice is also limited in place, i.e., it can only be practiced at Beit Mikdash (Deut. 12:13, 26) in Jerusalem. It is also limiting in the sense that only Kohanim (High Priests) were allowed to officiate as priests for the sacrifices. Prayer and repentance, on the other hand, can be done by anbody at anytime, anywhere (well, almost anywhere....filthy places or houses of idol worship, for instance, are no-nos). I would like to point something out with regards to the limitation of place. Leviticus 17:11 states that any blood sacrifice would need to be done on the altar [in Beit Mikdash]. Since Jesus' blood was never sprinkled on the altar, Jesus' death could not have been an act of universal atonement.
5) I'm sure this Biblical reference is going to shock some people, but according to Jeremiah (7:22-23), who was a prophet (i.e., G-d spoke through him), He never commanded us to perform sacrifices:
כִּי לֹא-דִבַּרְתִּי אֶת-אֲבוֹתֵיכֶם, וְלֹא צִוִּיתִים, בְּיוֹם הוציא (הוֹצִיאִי) אוֹתָם, מֵאֶרֶץ מִצְרָיִם--עַל-דִּבְרֵי עוֹלָה, וָזָבַח. כִּי אִם-אֶת-הַדָּבָר הַזֶּה צִוִּיתִי אוֹתָם לֵאמֹר, שִׁמְעוּ בְקוֹלִי--וְהָיִיתִי לָכֶם לֵאלֹהִים, וְאַתֶּם תִּהְיוּ-לִי לְעָם; וַהֲלַכְתֶּם, בְּכָל-הַדֶּרֶךְ אֲשֶׁר אֲצַוֶּה אֶתְכֶם, לְמַעַן, יִיטַב לָכֶם.
For when I brought your forefathers out of Egypt and spoke to them, I did not just give them commands about burnt offerings and sacrifices, but I gave them this command: Obey me, and I will be your God and you will be my people. Walk in all the ways I command you, that it may go well with you.
6) Vicarious atonement does not exist in Judaism. Moses tried this after the Golden Calf incident (Exodus 32:32-35), and HaShem said that each person is responsible for their own sin. Upon reading Ezekiel 18, it is quite clear that each individual is responsible for his own transgressions, as it is said, "the righteousness of the righteous shall be upon them and the wickedness of the wicked person shall be upon them (ver. 20)." Jeremiah (31:30) said that each man will die for his own iniquity. Isaiah (55:7) and Ezekiel (18:21-23), both prophets, argued in the same vain. Plus, if you read II Chronicles 7:14, you will see that it says, "[I]f my people, who are called by my name, will humble themselves and pray and seek my face and turn from their wicked ways, then will I hear from heaven and will forgive their sin and will heal their land." Notice how there is nothing about needing to make a sacrifice--teshuva more than suffices!
7) Atonement was not solely obtained through sacrifices, whether that would be blood, animals, or other objects. Prayer and repentance have very much have been a part of Jewish practice since Biblical times. We learn this from the Golden Calf incident, as well as from the Books of Jonah (3:10) and Esther (9:3). When we look at the story of King David and Batsheva, Nathan points out (2 Samuel 12:13) that as soon as David made his statement of repentance, HaShem forgave him. What these examples illustrate is that atonement was acquired without a drop of blood.
8) When the Jews are in exile, we cannot rely on sacrifices for two reasons. The first is because, like I previously explained, the only place that one can perform a sacrifice is in the Temple. We have not had one for nearly two millennia. The second reason comes from Hosea (3:4-5) when he prophesied that the Jews would be in exile for quite some time before the Messianic era, they would be without the sacrificial system. Furthermore, passages such as I Kings 8:44-52 and Jeremiah 29:12-14 inform us that without a Temple, our prayers take the place of sacrifices. Hosea 14:3 is also a verse that illustrates that prayer replaced sacrifices, but I want to reflect on this one momentarily because there is an important grammatical nuance that Christians mistranslate:
קְחוּ עִמָּכֶם דְּבָרִים, וְשׁוּבוּ אֶל-יְהוָה; אִמְרוּ אֵלָיו, כָּל-תִּשָּׂא עָוֹן וְקַח-טוֹב, וּנְשַׁלְּמָה פָרִים, שְׂפָתֵינוּ.
Take words with you and return to HaShem; say to Him, 'May You forgive all iniquity and accept good [intentions], and let our lips substitute for bulls [own emphasis added]."
That phrase I emphasized, "let our lips substitute for bulls," is important because when Christians translate it, they mistranslate it as "that we may offer the fruit of our lips." The reason why this is important is that פרי (fruit) is very similar to פר (bull). The problem is that the plural for פרי refers to fruit, is פריות, not פָרִים because פרי is a feminine word, whereas פר is masculine. [For those of you who don't know, the Hebrew nouns have two plural endings: ות- is for feminine nouns, and ים- for masculine nouns] The grammatical rules dictate that the proper translation of פָרִים is indeed "bulls." Plus, from a contextual standpoint, the only time that [the first] fruit was sacrificed was during Sukkot, which is a holiday of thanksgiving, not one of atonement. Although this grammatical nuance seems insignificant, Christians actually have a stake in this mistranslation. When mistranslated as "fruit," it hides the fact the real meaning of the text, which is that prayer substitutes sacrifice during the exile period. What this means is that Jesus' supposed vicarious atonement cannot cover all of our sins because HaShem already stated that prayer and repentance are the path of atonement during this specific time period (i.e., the exile of the Jewish people, also known as 70 C.E. to present).
Conclusion: Based on Biblical analysis what the Tanach tells us about atonement, there is no way that Jesus could have possibly died for anybody's sins. Christians will go to their "New" Testament to find a citation, but you cannot use that to find this supposed fulfillment because the claim of Christianity is that Jesus' blood sacrifice fulfilled the Tanach. The fact that no such criterion exists in the Tanach already dismisses Christian claims. Plus, this claim goes against everything that the Tanach teaches us about repentance, forgiveness, and personal responsibility. Judaism is highly democratic in the sense that everybody has the ability to ask for His forgiveness and return to His ways. The ability to ask forgiveness for one's iniquities and make a resolute effort not to transgress again is very much engrained in Tanach. May people realize the wonder of teshuva so we can herald the coming of Moshiach!