This is brought over, in part, from a related overview question.
For he has somewhere spoken of the seventh day in this way: “And God rested on the seventh day from all his works. - Hebrews 4:4
Hebrews 4 defines this rest that is being spoken of as that rest wherein God rested from all his works. Verses 3 and 5 further define that rest as a possession of God: My rest. It is that rest with which God has rested. In the creative act God accomplished all that He intended including those things which, in our temporal economy, appear as yet future. In this way the coming forth of Israel's ruler (yet future) is said to be "from of old, from ancient days". (Micah 5:2). In this way also the Lamb was slain from the foundation of the world (Revelation 13:8). God has rested and yet His works continue to unfold.
For good news came to us just as to them, but the message they heard did not benefit them, because they were not united by faith with
those who listened. For we who have believed enter that rest - Hebrews
Hebrews clearly presents faith/belief in "the message" as the mechanism of entry. In context it is the people of Israel's entry into the promised land that is treated immediately and "the message" is that God has promised to give them the land and has promised to go before them preparing the way and fighting the battles for them. In chapter 3 we see the flip side where unbelief is categorically given as the reason some did not enter rest:
And to whom did he swear that they would not enter his rest, but to
those who were disobedient? So we see that they were unable to enter
because of unbelief. - Hebrews 3:18-19
The exodus generation refused to believe "the message" regarding entry into what God had promised and they spun 'round in the wilderness for 40 years and most died there because of unbelief. The current generation, to whom Hebrews is written, are similarly in danger of falling short of belief in "the message" regarding the Christ of God. The entire epistle appeals to them to cease going about (through religious activities and works of the Law) to establish their own righteousness and to instead receive the righteousness of God which is by faith.
Take care, brothers, lest there be in any of you an evil, unbelieving
heart, leading you to fall away from the living God. But exhort one
another every day, as long as it is called “today,” that none of you
may be hardened by the deceitfulness of sin. For we have come to share
in Christ, if indeed we hold our original confidence firm to the end.
It appears as though it were possible to become hardened by sin's deceitfulness into developing an evil, unbelieving heart which may lead one to fall away from the living God. It is unclear in the text whether this falling away can occur once entry into "rest" has taken place or if developing unbelief simply hinders entry. Significantly, this passage links entering God's rest with "sharing in Christ" and also links the belief needed to enter with perseverance in that belief until the end.
Note: Theological bunny trails may begin here regarding perseverance as an act of human will, a gift of Divine grace, or some mixture of the two but they need not be entertained under this question.
So then, there remains a Sabbath rest for the people of God, for
whoever has entered God's rest has also rested from his works as God
did from his. - Hebrews 4:9-10
The content of the Sabbath rest that is available for the people of God to enter is not a list of prohibitions or prescriptions; it is a cessation of works. Just as God rests in this Sabbath from His works so too, we who enter into this Sabbath rest from our works.
The report from the scouts into the promised land (Numbers 13) described all the ways that the people of Israel could not possibly enter the land and overcome the difficulties they perceived there. In other words, their faith lay in what works they could accomplish rather than trusting in the promise of the Living God. They were not resting from their works, they were relying on them. This is the unbelief which hindered their entry: Trusting in their works rather than God's.
In like manner we must not entertain all the myriad ways that we cannot possibly enter into eternal life. We must not allow ourselves an evil, unbelieving heart by which we abandon full trust in the promise of God and by which we begin to try to work our way into the kingdom or count ourselves as worthy or not by virtue of our works. God, in Christ, has done all that can be or need be done regarding reconciliation and all who enter by Him will be rested.
For Israel, entrance into the rest of God necessitated trusting in the "work" that God would do on their behalf in the promised land. In the Gospel age entrance into the Sabbath rest now is entirely a matter of trusting that God will accomplish what He has promised in Christ. Once entered that person has ceased from his own works.
Resting from our works is not a condition for entry into God's Sabbath, it is a description of that rest. It is also not a motionless rest:
For we are his workmanship, created in Christ Jesus for good works,
which God prepared beforehand, that we should walk in them. -
And we also see Divine activity within this Sabbath rest of God:
And this was why the Jews were persecuting Jesus, because he was doing
these things on the Sabbath. But Jesus answered them, “My Father is
working until now, and I am working.” - John 5:16-17
In the following verses of John 5 Jesus explains how that He only does what He sees the Father doing. He has just done a work of healing on the legal Sabbath (religious rest), all of which fall within the confines of the Divine Sabbath (God's rest), to demonstrate that He is Lord of the Sabbath and that good works are always permissible because they are not man's works but God's works which, in the Divine economy, are already completed.
Jesus, in doing only God's work, was "resting" from his own works. Now we, who have believed in His Name and been saved are "in Christ" and have entered into the Sabbath rest of God with Him.. Jesus is the Sabbath rest that we may enter.
Come to me, all who labor and are heavy laden, and I will give you
rest. Take my yoke upon you, and learn from me, for I am gentle and
lowly in heart, and you will find rest for your souls. For my yoke is
easy, and my burden is light.” - Matthew 11:28-30