Calvin, among his other points, includes the point that Atonement is Limited; i.e., that Christ's death was sufficient for all but only effective for the elect.
What is the Biblical basis for this doctrine?
Limited atonement is defined in the Canons of Dort:
The Second Main Point of Doctrine: Christ's Death and Human Redemption Through It
Article 3: The Infinite Value of Christ’s Death
This death of God’s Son is the only and entirely complete sacrifice and satisfaction for sins; it is of infinite value and worth, more than sufficient to atone for the sins of the whole world.
Article 7: Faith God’s Gift
But all who genuinely believe and are delivered and saved by Christ’s death from their sins and from destruction receive this favor solely from God’s grace—which God owes to no one—given to them in Christ from eternity.
Article 8: The Saving Effectiveness of Christ’s Death
For it was the entirely free plan and very gracious will and intention of God the Father that the enlivening and saving effectiveness of his Son’s costly death should work itself out in all the elect, in order that God might grant justifying faith to them only and thereby lead them without fail to salvation. In other words, it was God’s will that Christ through the blood of the cross (by which he confirmed the new covenant) should effectively redeem from every people, tribe, nation, and language all those and only those who were chosen from eternity to salvation and given to him by the Father; that Christ should grant them faith (which, like the Holy Spirit’s other saving gifts, he acquired for them by his death). It was also God’s will that Christ should cleanse them by his blood from all their sins, both original and actual, whether committed before or after their coming to faith; that he should faithfully preserve them to the very end; and that he should finally present them to himself, a glorious people, without spot or wrinkle.
Article 9: The Fulfillment of God’s Plan
This plan, arising out of God’s eternal love for the elect, from the beginning of the world to the present time has been powerfully carried out and will also be carried out in the future, the gates of hell seeking vainly to prevail against it. As a result, the elect are gathered into one, all in their own time, and there is always a church of believers founded on Christ’s blood, a church which steadfastly loves, persistently worships, and here and in all eternity praises him as her Savior who laid down his life for her on the cross, as a bridegroom for his bride.
Rejection of the Errors
Having set forth the orthodox teaching, the Synod rejects the errors of those
Who teach that God the Father appointed his Son to death on the cross without a fixed and definite plan to save anyone by name, so that the necessity, usefulness, and worth of what Christ’s death obtained could have stood intact and altogether perfect, complete and whole, even if the redemption that was obtained had never in actual fact been applied to any individual.
Who teach that the purpose of Christ’s death was not to establish in actual fact a new covenant of grace by his blood, but only to acquire for the Father the mere right to enter once more into a covenant with humanity, whether of grace or of works.
I have quoted three articles and a rejection of two errors, but for a fuller definition of limited atonement, click the link and read all nine articles and a rejection of seven errors.
You can see above, in article 3 juxtaposed with articles 7-8, that the atonement is "sufficient to atone for the sins of the whole world" but that the cross "should effectively redeem from every people, tribe, nation, and language all those and only those who were chosen from eternity to salvation and given to him by the Father." This has been summarized by many apologists as, "sufficient for all, efficient for only the elect."
Therefore the atonement is not "limited" in its power, but in its purpose. It's common for opponents of reformed theology to advance the argument that a limited atonement means the cross wasn't "powerful" enough to achieve the purpose of the salvation of every person; that's not the case. God's purpose was to save the elect, and that purpose was accomplished on the cross.
To better capture the fact that the atonement's purpose was to redeem a "particular" people and that it is "definite" in its scope, some have begun calling it by other names:
Now for the meat.
It's all throughout the Gospel of John that Christ died for his "people" or his "sheep" and not for everyone else:
37 Everyone whom the Father gives me will come to me, and the one who comes to me I will never send away. ... 39 Now this is the will of the one who sent me—that I should not lose one person of every one he has given me, but raise them all up at the last day.
64 But there are some of you who do not believe. ... 65 Because of this I told you that no one can come to me unless the Father has allowed him to come.
John 10: The Good Shepherd
11 I am the good shepherd. The good shepherd lays down his life for the sheep. ... 14 I am the good shepherd. I know my own and my own know me. ... 26 But you refuse to believe because you are not my sheep. 27 My sheep listen to my voice, and I know them, and they follow me. 28 I give them eternal life, and they will never perish; no one will snatch them from my hand.
John 17: Jesus Prays to the Father
6 I have revealed your name to the men you gave me out of the world. They belonged to you, and you gave them to me, and they have obeyed your word. ... 9 I am praying on behalf of them. I am not praying on behalf of the world, but on behalf of those you have given me, because they belong to you.
In Matthew 1:21 we see the same idea, of God saving "his people":
She will give birth to a son and you will name him Jesus, because he will save his people from their sins.
God's chosen people is Israel, or the church. In reformed theology, a continuity between the two is recognized, so that the "true Israel" and the "invisible church" is the same as the "elect." The scriptural support for this idea is beyond the scope of the current question, but it's worth mentioning since Peter said in Acts 5:31 that Christ died for Israel:
God exalted him to his right hand as Leader and Savior, to give repentance to Israel and forgiveness of sins.
It is also the teaching of Paul that Christ died for "the church":
Watch out for yourselves and for all the flock of which the Holy Spirit has made you overseers, to shepherd the church of God that he obtained with the blood of his own Son.
Husbands, love your wives just as Christ loved the church and gave himself for her.
This is consistent with the song the saints sing in Revelation 5:9-10:
At the cost of your own blood you have purchased for God persons from every tribe, language, people, and nation. You have appointed them as a kingdom and priests to serve our God, and they will reign on the earth.
John Owen contends in his classic defense of limited atonement The Death of Death in the Death of Christ that Romans 8 supports the doctrine:
Who is the one who will condemn? Christ is the one who died (and more than that, he was raised), who is at the right hand of God, and who also is interceding for us.
Owen's Commentary in The Death of Death
That he died for all and intercedeth for some will scarcely be squared to this text, especially considering the foundation of all this, which is (verse 32) that love of God which moved him to give up Christ to death for us all; upon which the apostle infers a kind of impossibility in not giving us all good things in him; which how it can be reconciled with their opinion who affirm that he gave his Son for millions to whom he will give neither grace nor glory, I cannot see.
There is one other point to consider. I could have titled this section "indirect Biblical evidence." The fact is that rejecting limited atonement would make for an inconsistent system of theology. The following two quotes should demonstrate what I mean.
A. A. Hodge says in his book The Atonement:
Every form which it is possible for the General Atonement Theory to assume necessarily involves the hypothesis that in its essential nature the Atonement effects only the removal of legal obstacles out of the way of the salvation of men, making God reconcilable, not actually reconciling him; making the salvation of all men possible, not actually saving any. But the Scriptures teach that Christ actually came to save those for whom he died ... Again the Scriptures decleare that the effect of Christ's death is reconciliation and justification. ... The design of Christ, moreover, was to secure for those whom he died the direct effect of remission of sins, peace with God, and deliverance from the curse of the law, from wrath, from death, from sin, &c. ... But to make salvation possible, to make possible purification, deliverance, reconciliation, is something very different indeed from actually saving, purifying, delivering or reconciling. No man has the right to empty the glorious terms in which the gospel is revealed of all their saving power.
Hodge cites Matthew 18:11; Luke 19:10; Romans 5:10; 2 Corinthians 5:21; Galatians 1:4, 3:13, and 4:5; Ephesians 2:14-16; 1 Thessalonians 1:10; 1 Timothy 1:15; Hebrews 2:14; and 1 Peter 1:18.
Robert Morley similarly said in his book Studies in the Atonement:
When Christ lived, died, was buried, arose, ascended, and sat down at the right hand of the Majesty on high, we are told that the ones for whom He did these things are to be viewed as being in such a life union with Him as their covenant head and representative that it is said that they lived, died, were buried, arose, ascended and sat down at the Father's side 'in Christ' (Rom. 6:1-11; Gal. 2:20; 6:14; Eph. 2:5-6).
To say that Christ died for all is to say that all died in Christ. It means that unbelievers are to be told that they have been crucified with Christ, been buried with Christ, have been resurrected with Christ and have ascended and sat down with Christ. This position is so manifestly false that it should grieve the child of God even to consider it.
For more on the "consistency" point, visit the link to the Canons of Dort and scroll to "Rejection of the Errors" under the second main point.
The Bible doesn't really say that Jesus' blood is sufficient for all, but it strongly suggests it. Jesus is the perfect lamb, and He give the power to become a son of God to " as many as received him." Surely the Lord God's blood is more than sufficient for the atonement of anyone He wishes to atone.
However, in regards to the Blood's efficacy, this is a matter of the election of God. It is not to say that the saving power of the Blood simply doesn't work for non-elect. That doesn't make sense. Jesus simply shed His blood for his elect (this is not to say that this was a simple process). This is probably what Calvin is talking about, since Unconditional Election is the basis for Limited Atonement. When Jesus took the cup he said,
"For this is my blood of the new testament, which is shed for many for the remission of sins." Matt. 26:28
Just like "many are called." Jesus' blood is "shed for many." Not all.