Do you think you have more endurance than God? Certainly not! We wouldn't make the claim that we have more stamina than our Creator. By not having a day of rest we dishonor God. How? Because He made the day holy.
In Exodus 20, God gives Moses the 10 commandments on Mount Sinai. One of those commandments was to rest. I personally believe that this was in response to the slavery they faced back in Egypt. God delivered them from slavery and forced labor, and rewarded them with the gift they needed most: Rest! God gave additional details in this commandment. He wanted to make it clear that the Sabbath was for everyone. The Sabbath was for family members, servants, animals, and foreigners. The human body isn't meant to run without rest. That is why God allowed mankind to create robots. You and I are not robots, we need rest!
I am the poster child for not taking a break until I am physically and emotionally exhausted. I don't know why I try to be everything to everyone and neglect resting myself. God created the Sabbath to keep us from burning ourselves out.
What should be done on the holy day? Resting, worshiping, honoring God with the time he has given you to rest, and giving others the opportunity to rest. Remember in the original commandment, not only does it say we should rest, but we should also make sure other people around us are resting.
Well Paul tells us not to let anyone judge how we spend the Sabbath. See Colossians 2:16-17. This doesn't mean we shouldn't honor the Sabbath. I think it means don't let those around us shame us because we do honor the holy day.
There remains, then, a Sabbath-rest for the people of God; for anyone who enters God’s rest also rests from their works, just as God did from his. Let us, therefore, make every effort to enter that rest, so that no one will perish by following their example of disobedience. Hebrews 4:9-11 (NIV)
Then he said to them, “The Sabbath was made for man, not man for the Sabbath. So the Son of Man is Lord even of the Sabbath.” Mark 2:27-28 (NIV)