Be imitators of me, Paul writes in I Corinthians 11:1, just as I also am of Christ. The greater context of this verse is perhaps summarized more in the words of Jesus, than in the words of Paul. Jesus said, the Son of Man has come to seek and save that which is lost (Luke 19:10). This entire section in Corinthians, going back at least to chapter 9, is primarily about seeking the lost with the ultimate goal of their salvation. Paul wrote in 9:22b I have become all things (a Jew, under the Law, without the Law, weak) to all men, that I may by all means save some. I believe that what Paul meant was that he engaged these different peoples right where they were - speaking their language - sharing in their cultures and customs. Paul identified with all sorts of people with the aim of sharing the gospel and seeing some come to Christ.
Paul was obviously not the first missionary to use this method. It was the method of Jesus, Himself.
Jesus took on the flesh of humanity, was tempted, and suffered in order to become a merciful and faithful high priest, and to make propitiation for our sins, and come to the aid of us who are tempted (Hebrews 2:14-18).
Jesus came to seek and save that which was lost. He came seeking. He sought the lost at wedding feasts where participants drank wine. He sought the lost when he approached tax collectors, whores, prostitutes, fishermen, businessmen, adulterers... He sought the lost in the discounted diseased. And yes, he sought the lost in synagogues and in debates with religious leaders.
I believe Jesus' approach was the model for Paul's approach and should be our model as well - to be with people, right where they are (though not partaking in their sins) and to be for the good of people and their ultimate salvation. We should be all things to all people in the same manner that Jesus was. That means that we engage people where they are - at wedding feasts drinking wine, at tax collectors houses, and dining with sinners. It means that we love prostitutes and adulterers, fishermen and businessmen - we love them all for their good and for the sake of the gospel.
Being missional means being relevant - but not for the sake relevance - for the sake of the gospel, and Christ is our model - becoming all things to all men that he might save some.
I do not ask You to take them out of the world, but to keep them from the evil one. As You sent me into the world, I also have sent them into the world. John 17:15,18