Paul is not saying that our physical bodies don’t need rest. He is saying that we need to be Spiritually alert at all times. We are to be alert, sober, and dressed in our battle armor each moment of every day. There is a Spiritual battle being waged for the control of our minds 24/7/365. We may not see the combatants but we can know from Scripture that the battle is being waged continuously. There are examples of this in the following texts as well as other places:
12 For we do not wrestle against flesh and blood, but against the rulers, against the authorities, against the cosmic powers over this present darkness, against the spiritual forces of evil in the heavenly places.–Ephesians 6:12 (ESV)
The prince of the kingdom of Persia withstood me twenty-one days, but Michael, one of the chief princes, came to help me, for I was left there with the kings of Persia,–Daniel 10:12 (ESV)
We just need to remember that if God is for us, who can defeat us? So, we encourage each other with these words.