It is important to remember that Paul's primary concern in the book of Ephesians is dealing with the Gentile believers in relation to the Jews. Often we become so concerned with this verse or that verse that we forget Paul's greater discussion.
The big question when considering this verse is "what were we far away from"? The answer is in the last verse: we were far away from God and His covenant people. The language is verse 12 is not merely talking about separation from God, but also separation from Israel. To Paul, these are two sides of the same coin. To be separate from God's people is to be separate from God's promises, and thus from God Himself.
But the turning point is Christ. Through Christ, we, the Gentiles, have been made citizens of God's people. To am not talking about replacement theology here, where the Church has replaced Israel. I mean that we have become part of Israel through the blood of Jesus Christ. The redemption that was wrought through Christ's atonement has given us access to the promises of Abraham.
Remember that we do not have a right to salvation. We have not merited it, nor do we have some claim to it by birthright. Indeed, as Gentiles, we don't even have the same claims that the ancient Jews had. Do not take it for granted.
It is a gift! A great gift: undeserved and unasked for, yet here! We need to be focused in life on the gratitude we have to Jesus for what He has done for us. Think that out this week, and relish the joy of being close to God.