The Bible Says
Romans 11:7-8 Meaning

Even though the Israelites say they are being zealous for God and want to prove they are righteous through following the rules of the Law, they have failed (Romans 10:2). What then? What Israel is seeking, it has not obtained (v 7). Israel has been seeking righteousness through the Law, but has not obtained righteousness through that means. 

This makes intuitive sense; following a list of rules to "make God favor me" is tantamount to treating God like a genie in a bottle. Trying to make God do something for us could never cause us to be transformed by God. On the other hand, those whom God set aside, as a remnant (v 5), have obtained what the rest of Israel seeks, those who were chosen obtained it (v 7): through faith, not by following rules from the Law (Romans 9:32). 

This also makes intuitive sense; believing and following God leads us to become who God wants us to be. God desires us to be righteous. "Righteous" simply means to live in harmony with the way things are supposed to be. To live consistent with God's (good) design for the world. God made the world perfect, pronounced it "good," and then it fell because of sin. 

When we walk in "righteousness" it simply means we choose to live in harmony with others according to God's design for us. We do this because we have faith that God knows what is best for us. Paul will describe to his Roman audience what this looks like in chapters 12-16. Living this way can only be accomplished by faith, believing God in our daily living that His ways are for our best.

The righteous remnant includes Paul, who at the beginning of the chapter showed that he, an Israelite, is also a believer in Jesus Christ. This definitively proves that Paul is not teaching that God has rejected those of Israel, for then Paul would be teaching that he himself has been rejected.

The rest of Israel who are not the remnant, those who have pursued following deeds of the Law, have failed to believe God and follow God's will. Thus Paul says and the rest were hardened (v 7) because of their unbelief. And though the Jews (think that they) seek righteousness, they have not obtained it. 

Here Paul quotes several Old Testament verses (Isaiah 29:10, Deuteronomy 29:4) to show that though God is not rejecting the Israelites completely, He is withholding understanding from them, giving them a period of disconnection from Him: Just as it is written, "God gave them a spirit of stupor, eyes to see not and ears to hear not, down to this very day (v 8)." Just as God further hardened Pharaoh's heart after his initial rejection of God's commands, so it seems God is doing also to Israel (Exodus 8:15, 9:7, 34-35, 10:1). 

Paul will tell us soon that this hardening will result in a great blessing for the world. Because Israel rejected Jesus as Messiah, many Gentiles came to faith and will be grafted in to the olive tree that is spiritual Israel. Similar to how God will use all things to work together for the good of all believers (Romans 8:28-29) God will use all things to work for good for His people Israel.