Is baptism by water necessary for salvation? John 3:5  Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God. 

If the Kingdom of God is inside you (Luke 17:21), does this mean that one cannot have true inner peace with God unless he is baptized by both water and the Holy Spirit?

And does this mean that a person cannot receive salvation into the Kingdom of Heaven without baptism in both water and the Spirit first?
No, it is not necessary for salvation. Either Christ's work is complete at the cross or it is not. 

Lets go to scripture.

Act 11:16  Then remembered I the word of the Lord, how that he said, John indeed baptized with water; but ye shall be baptized with the Holy Ghost. 
Water Baptism is merely a symbol of the Holy Spirit which baptizes.

No man made tradition provides salvation.

Also , Apostle Paul when he was In prison.
Act 16:30  And brought them out, and said, Sirs, what must I do to be saved? 
Act 16:31  And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house. 

He didn't tell him, go get baptised and then you will be saved.