On the Day of Atonement, God requires that absolutely no work be performed (Leviticus 16:29; 23:28-31; Numbers 29:7), symbolizing that human effort is completely useless in making the proper atonement needed to keep living after sin. The Israelites could do nothing but observe what occurred at the Tabernacle, watching as the young goat was led away with all their sins. Likewise, we can do absolutely nothing to add to Christ's atoning work. Thus, it is a day without work for us as well.
Israel's works nearly condemned the nation to obliteration. In particular, the Golden Calf was a work of Aaron's hands (Exodus 32:4-5). No matter how he tried to pass it off, he deliberately fashioned an idol out of gold, something he had to work at. Similarly, the work of Nadab's and Abihu's hands included offering profane fire (Leviticus 10:1). In Haggai 2:14, God remarks on Israel's spoiling of everything she puts her hands to: “'So is this people, and so is this nation before Me,' says the LORD, 'and so is every work of their hands; and what they offer there is unclean.'” The works of men always contain defilement, so on the day when God removes the filth, no work can be done, lest more corruption be introduced.
The only work permitted on the Day of Atonement was performed by the high priest and by the man who led the azazel away, and both had to have an atonement made for them. For us, it is a day of solemn remembrance of the perfect work of our High Priest, who gave us precious access to the Father and removed our sins.
Atonement is also a day of afflicting one's soul. This requirement could serve as a reminder of the fasting Moses did during his interactions with God. There is overwhelming gravity in all that was involved when he fasted for forty days on back-to-back-to-back occasions. Two of those times involved meeting directly with God, receiving a pattern for life from His incomparable mind. The middle period of fasting reflects how seriously God regarded the sins and the enormity of what was at stake due to Aaron's and the nation's transgressions.
