Upon hearing the phrase "sacrificial system" in the modern world, many of our minds automatically think of barbarism and primitivism. Some in the Jewish world impatiently await a Messianic era with the restoration of the sacrificial system (note the seventeenth blessing of the Amidah). Others would go as far as saying that a return to the sacrificial system would be a regression in Jewish spiritual development.
I don't feel bad questioning this practice, not simply because questioning is a part of Jewish thought, but because this debate has been going on for centuries. Maimonides thought that the Israelites were eventually supposed to wean themselves off the sacrificial system (Guide for the Perplexed, III, xxxii), and on the other side, Nachmanides opined that G-d desired our sacrifices. Even in the ancient times, the prophets had this debate. Samuel (I Samuel 15:12) asked if G-d delighted in burnt offerings. Isaiah (1:2) also questions it. In response to how to go about repentance, Hosea (6:6) said that G-d desires loving-kindness, not sacrifices. Jeremiah (7:22-23) made as radical statement as G-d never commanded the sacrificial system in the first place.
Aside from its occasional mentioning in the siddur and codification in the Mishne Torah, the sacrificial system plays no practical role in our daily lives. Why? Because, as Maimonides points out, there is no Temple. No Beit Mikdash means no sacrifices. That is why when the Second Temple fell in 70 CE, the Jews faced a spiritual, existential question: Can Judaism survive without the sacrificial system?
When looking at this topic, we are mislead in one of our most essential premises in this discussion, that being the very notion of sacrifice. The word sacrifice is of Greek origin (thusia). Sacrifices were meant to give something up in order to placate the pagan deities. What we see in contradistinction is that in Judaism, the word that is commonly mistranslated as "sacrifice," קרבן, comes from the root קרב, which means "to draw close." Rather than appeasing to the Infinite One, like Nachmanides thought, the sacrificial system was a way to draw closer to G-d. During the time of Leviticus, the universal form of worship was making sacrifices. In order to evade idolatry, the idolatrous elements, such as human sacrifice or praying towards the east to teach others about sun worship. Much to the dismay of right-winged Christians and certain Orthodox Jews, sacrifices were a means, not an ends to spiritual practice. To quote the late prophet Micah (6:8), "What does Hashem require of you? To do justice, to love mercy, and to walk humbly with Him."
To go to the previous question, what happened to Judaism when the Temple was destroyed? If the sacrificial system was the only way to get closer to G-d, then the Jews should have been extinct a little after the Temple's destruction, rather than still surviving today. As Sir Chief Rabbi Jonathan Sacks so eloquently stated, the Jews had to innovate in the midst of their spiritual angst. Rather than cling to their past by obsessing themselves with the non-existent sacrificial system, the Jews came up with other practices that brought the individual Jew the same closeness to G-d that one felt during the times when either the First or Second Temple was standing.
As Rabbi Sacks tells us, the Jews of yore had to perish or adapt. They opted for the latter. They chose to partake in acts of loving-kindness (gemilut chasadim) to "know G-d in His ways." Jews chose Torah study so we could have an idea of what it is like to have G-d speak to us. Our prayers take the place of sacrifice because we quite literally let our words do the talking of how we feel. Our hospitality (i.e., our table) substituted the altar that once atoned for the people Israel.
Sacrifices are no longer practiced. However, by reading Leviticus and how they came closer to G-d, we connect to the past through Torah study and understanding what their modus operandi was. We also connect to the present in the sense that no matter what time period it is, the individual will have the yearning to become closer to G-d, and that our innovative practices are just as good at doing the job, and I would certainly argue more so, as the Levitical sacrifices.