Remember what made the religious leaders of Jesus' day so angry with him? Of course, it was because they realized he was claiming essential deity in his perfect unity with God the Father. Millions have missed this over the centuries, but these Jews did not. And they were right about one thing - this claim was either blasphemy or it was true. They landed on the wrong conclusion there.
But there was a particular implication and expression of this claim. And it is another reason why Jesus infuriated them so much. He kept doing stuff on the Sabbath. Stuff they didn't think he should do. Like healing people (Mt 12:9; Lk 13:10-16; Jn 5:1-14; 9:1-13), letting his disciples eat (Mt 12:1-2) and commanding someone else to "work" (Jn 5:8-10).
In that last example, John records explicitly that the religious leaders were "persecuting Jesus because he was doing these things on the Sabbath" (v.16). How did Jesus respond to that? "My Father is still working, and I am working also" (v.17). They realized that in this statement Jesus was claiming ontological unity with the Father (v.18). But they missed - or rejected - what he was communicating about the Sabbath.
In his creation week, God established the Sabbath (Gn 2:2-3). It was his idea. Then he instituted it as a weekly observation for his people, a sort of creation week rehearsal (Ex 16:23-30). He further communicated how big a deal this observance was, and he explained the function of this sign: "...so that you will know that I am the LORD who consecrates you" (Ex 31:12-17).
By the time Jesus had hit the scene, the Jewish rulers were all over the observance of the covenant ritual - in fact, they cranked up the standards for what constituted "work" - but they totally missed the sign. This is a key theme in John's Gospel. The controversy in ch.5 over the healing of the lame man comes within what many call John's Book of Signs (chs. 2-12).
These who were supposed to be the religious leaders among God's people were plodding along with their eyes to the ground, fixated on the furrows of legalism, as though they could somehow cultivate in their own fallen humanity a crop that would yield their righteousness before God. Now God himself stood before them on the Sabbath, telling them he was "still working." They should have considered what that must mean if it were true.
The weekly Sabbath modeled in creation and mediated through Moses was a sign. A sign is not the reality. It is what points to the reality. If YHWH had made such a big deal about the Sabbath, and now Jesus was claiming to be YHWH working on the Sabbath, then to what reality was the sign pointing? What was Jesus doing?
He was restoring. He was bringing life. He was rehearsing the creation week. Over and over in John's Gospel Jesus is equated with ζωή (zoe) - eternal life, true life. The Fall had ruined God's original creation, so now he was working in a new creative week. The weekly Sabbath was pointing to an ultimate Sabbath that was not yet realized. God's re-creative work would come to completion in Jesus through his own death, burial and resurrection. We symbolize the true Sabbath every time someone comes out of the baptismal waters. With Christ's death he declares about the work, "It is finished." With his resurrection comes the true Sabbath rest, eternal life.
Remember, the LORD said the Sabbath was "so that you will know that I am the Lord who consecrates you." We are consecrated - set apart as holy, set apart for life - by belief in Jesus, the Christ, the Eternal Son who is our zoe. That eternal life is the true Sabbath, our true rest through belief in Jesus (Heb 3:7-4:11).
Don't fixate on the sign, or worse yet, miss it altogether. How ridiculous that anyone would vilify the LORD for doing his restorative work in ways or at times we think he ought not - he is the Lord of the Sabbath! But we can be just like the blind Pharisees, only finding fault when God is working, denying his lordship and missing what is important. Let Christ lift your eyes to the reality to which the sign points, to him who is our life.
The Father is still working - and so should we be. He is still consecrating people through the witness of his Church. The true Sabbath comes through belief in the Son's work. He came so that we might have life in his name (Jn 20:31). That is our Sabbath. That is our rest.