Were the Christians meant to be of a society all of their own?
The Church, i.e., "Christians," are the "set-apart ones." the very nature of what we are makes us separate from the remainder of humanity.
If ye were of the world, the world would love his own: but because ye
are not of the world, but I have chosen you out of the world,
therefore the world hateth you. 20 Remember the word that I said unto
you, The servant is not greater than his lord. If they have persecuted
me, they will also persecute you; if they have kept my saying, they
will keep yours also. —John 15:19-20 (AV)
If we (believers/Christians) are not of the world, then we are, indeed, separate from the world. Ours is a spiritual society (as mentioned by Aaron Korn above).
I have given them thy word; and the world hath hated them, because
they are not of the world, even as I am not of the world. 15 I pray
not that thou shouldest take them out of the world, but that thou
shouldest keep them from the evil. 16 They are not of the world, even
as I am not of the world. —John 17:14-16 (AV)
The words of Yeshua are both powerful and definitive, "You are not of the world..." Though you are in the world, you are not a part of the world system, you have been separated out from that.
Wherefore come out from among them, and be ye separate, saith the
Lord, and touch not the unclean thing; and I will receive you, —2
Corinthians 6:17 (AV)
Here, Paul cited a portion of Isaiah 52:11 and Ezekiel 20:41. Those passages speak of Israel’s redemption. God’s people are redeemed from pagan bondage (come out from them and be separate) in order to be clean before God (touch no unclean thing) and thus enjoy fellowship with him (I will receive you).
In Galatians Paul spoke of the bondage of those who are under the obligation of the Law (Galatians 3:13–14; 4:5; cf. 2 Corinthians 3:7–9). Paul wrote to Titus that redemption implied two things: (a) deliverance “from all wickedness” and (b) a purified people who are “His very own, eager to do what is good” (Titus 2:14). Personal purity makes it possible to serve God and be received by him.
All of this comes together to show that Christians are a separate society within a society (in the world, but not of the world). Our being a separate society does not suggest elitism; where we have no interaction with the world around us. Yeshua most definitely interacted with the world around him, but he was never a part of the world system.