The whole chapter expresses the need to abandon works with a view to earning salvation. Beginning with a story about OT peoples who had opportunity to enter a physical rest - via the Sabbath and the promised Land, the same principle plays out in NT with the Spirit of God now active in His people, and a focus on Jesus, instead of law.
The preceding chapters speak of the centrality of Jesus and his role as High priest, his sacrifice and the benefits that flow from God's loving plan
let us fear if, while a promise remains of entering His rest, any one of you may seem to have come short of it... because it was not united by faith in those who heard.
3 For we who have believed enter that rest
and those who formerly had good news preached to them failed to enter because of disobedience, (and disbelief)
10 For the one who has entered His rest has himself also rested from his works, as God did from His.
11Therefore let us be diligent to enter that rest, so that no one will fall, through following the same example of disobedience. 12 For the word of God is living and active and sharper than any two-edged sword, and piercing as far as the division of soul and spirit, of both joints and marrow, and able to judge the thoughts and intentions of the heart.
No one can stand by keeping a law - God's word sees deeper within us than we do in our inflated and deceived view of ourselves as being somehow good enough. Jesus has accomplished all necessary for true life and in that life - rest - trust - peace - assurance...
14 Therefore, since we have a great high priest who has passed through the heavens, Jesus the Son of God, let us hold fast our confession. 15 For we do not have a high priest who cannot sympathize with our weaknesses, but One who has been tempted in all things as we are, yet without sin.
16 Therefore let us draw near with confidence to the throne of grace, so that we may receive mercy and find grace to help in time of need.
This has nothing to do with a 'day' or a 'Sabbath'. These are the shadows of what was always coming - true life only through Jesus the Christ with the Father. The Sabbath was a time of rest every week - God offers through Christ a rest that is permanent and everlasting - only by GRACE. (not by works as they are part of the law - the law and sin bring death)
What effort must one make to enter that rest?
To believe, to trust, to obey - not the law per se, but God's spirit dwelling in us as a deposit only in this life - trusting in what Jesus accomplished on the cross and the resounding victory won for all people.
It might be summed up by - 'seek first the kingdom'. Which is to do as Jesus always did - even at great cost and anguish, to put God's will before his own. Then, the rest will be ours forever.