When Paul wrote to Titus, he stopped one verse short of us having a 3:16. Instead, we'll look at Titus 1:16 (with verse 15 included for context).
15 To the pure, everything is pure, but to those who are defiled and unbelieving nothing is pure; in fact, both their mind and conscience are defiled. 16 They profess to know God, but they deny Him by their works. They are detestable, disobedient, and disqualified for any good work.— Titus 1:15-16
Paul had been to the island of Crete, and when he left, he left Titus there to continue the work of building the church on Crete. It was a tough mission field for Titus - there were people who wanted to overthrow the church, and were financially profiting from teaching what was wrong. (vv. 10-11) In fact, here's how one of their own described them.
12One of their very own prophets said,
Cretans are always liars, evil beasts, lazy gluttons.
— Titus 1:12
Titus certainly had his work cut out for him! But, Paul's advice to him on selecting leaders of this young church can help us today, even though most of us probably attend well-established churches (or could if we wanted to). Paul starts verse 15 by saying "To the pure, everything is pure..." Paul hasn't been talking about those who are pure, but he is reminding Titus that there are people who have accepted the Word of God in their hearts, and have been made pure. This is the type of person we should strive to be. Paul mentioned purity when he told the Philippian church where they should focus their thoughts.
8 Finally brothers, whatever is true, whatever is honorable, whatever is just, whatever is pure, whatever is lovely, whatever is commendable—if there is any moral excellence and if there is any praise—dwell on these things.— Philippians 4:8
Even Jesus mentioned those who are pure in His Sermon on the Mount.
8 Blessed are the pure in heart, because they will see God.— Matthew 5:8
Paul follows this short description with a contrast. The "defiled" are different; they claim to know God, but their words and their works are at odds with one another. They may know about God in their heads, but they do not know Him in their hearts. Paul describes these people as "detestable, disobedient, and disqualified for any good work." (v. 16)
Rather than focus on what these folks (or us today) have done wrong, think about this. If Paul had thought that the detestable, disobedient, vile beasts of Crete were beyond hope, would he have gone and started a church, and left Titus there to continue it? I doubt it. No matter how far gone we may think we are, it is God's grace that can bring us out of that, and purify our hearts. The purity that Paul spoke of in verse 15 is not something we can manufacture ourselves; rather, it is the outcome of us allowing God to come into our hearts and lives, and Him getting rid of the things that are keeping us from the purity He desires.
I pray that we will allow the Holy Spirit to purify our hearts, and make us willing, strong vessels for His work.