How can the Holy Spirit abide in our sinful bodies?

When we are born again, we receive the Holy Spirit. Our new spirit is 100% sinless, but our flesh is still 100% sinful. How can the Holy Spirit (God) take up residence in our body while it is full of sin? I thought God could not have sin in His presence.

Your question addresses deep theological ideas that are hard to understand, since the Bible gives us limited information on this subject. Nevertheless, we have some specific points we can make.

The Godhead may enter into the presence of sin without being compelled to judge it under all circumstances. The key to understanding why is found in knowing which member of the Godhead is in view. Throughout Scripture we find examples of each member of the Godhead interacting with sinful man, including the Lord speaking with Cain, appearing before Moses, making a covenant with Abraham, taking on human flesh to dwell among men, and indwelling believers bodies.

So obviously, God may interact with sinful men under some circumstances, and based on Scripture we see that the Second and Third members of the Godhead are the Persons capable of such interaction. On the other hand, God the Father is not. God the Father cannot be in the presence of sin without judging it, as He says:

Ex. 33:19 And He said, “I Myself will make all My goodness pass before you, and will proclaim the name of the Lord before you; and I will be gracious to whom I will be gracious, and will show compassion on whom I will show compassion.”
Ex. 33:20 But He said, “You cannot see My face, for no man can see Me and live!”

God’s “face” is euphemism for the Father’s full glory, so to see the “face" of God is to enter into the presence of the Father's full glory, which can only be witnessed by those who share His perfection. Sinful flesh must undergo judgment if brought into the presence of a holy God.

On the other hand, the Holy Spirit may indwell believers because though He is God, He is not the full glory of the Godhead. Similarly, Jesus may walk among sinful men because though He is God, Jesus is not the full glory of the Father. Hebrews says:

Heb. 1:3 And He is the radiance of His glory and the exact representation of His nature, and upholds all things by the word of His power. When He had made purification of sins, He sat down at the right hand of the Majesty on high,

Jesus is the “radiance” of the Father’s glory (i.e., an indirect way of witnessing the Father’s glory) and a “representation” of the Father’s nature (i.e., patterned after the Father). Jesus is no less God, but neither is He the Father.

So God the Spirit and God the Son may exist in the presence of sin without requiring judgment because neither by Himself is the full glory of the Father. Only God the Father remains unable to abide with sin, as John says: