Does predestination mean God prevents some from being saved?

We can certainly sympathize with your doubts over the Bible's teaching on predestination because most Christians share your experience as they wrestle with this difficult concept for the first time. Your concerns center on the prospect of a person who wants to be saved but is prevented by God's failure to choose them, but the scenario you proposed is impossible, according to Scripture. 

According to the Bible's teaching, there is no such thing as a person who "wants" to be saved but can't because God hasn't chosen that person. On the contrary, the Bible states plainly that no one (apart from God's grace) ever wants to be saved:

Rom. 3:10 as it is written, 
“THERE IS NONE RIGHTEOUS, NOT EVEN ONE;
Rom. 3:11  THERE IS NONE WHO UNDERSTANDS, 
THERE IS NONE WHO SEEKS FOR GOD;

No one seeks God on his own. The human condition from birth is set opposed to God and is an enemy of God, and there is no way in human terms that a person will ever overcome that nature and seek to be saved by Jesus. 

And the Bible also teaches that the call of the Gospel is foolishness to those who have not been elected by God to hear it:

1Cor. 1:17 For Christ did not send me to baptize, but to preach the gospel, not in cleverness of speech, so that the cross of Christ would not be made void.
1Cor. 1:18  For the word of the cross is foolishness to those who are perishing, but to us who are being saved it is the power of God.
1Cor. 1:19 For it is written, 
“I WILL DESTROY THE WISDOM OF THE WISE, 
AND THE CLEVERNESS OF THE CLEVER I WILL SET ASIDE.”

1Cor. 1:23 but we preach Christ crucified, to Jews a stumbling block and to Gentiles foolishness,
1Cor. 1:24 but to those who are the called, both Jews and Greeks, Christ the power of God and the wisdom of God.

Only those called by God (i.e., predestined) are able to hear and understand the Gospel so as to respond affirmatively. For everyone else, the Gospel is foolishness, the Bible says. God did not "prevent" someone from hearing the Gospel, because everyone is born unable to hear and respond already. Unbelief is the natural condition of all mankind from birth, so only the power of God (see 1 Corinthians 1:24) brings a person to saving faith. The Bible calls this moment being "born again", so we must be born again by God or else we would never believe. As Paul says:

1Cor. 1:30 But by His doing you are in Christ Jesus, who became to us wisdom from God, and righteousness and sanctification, and redemption,
1Cor. 1:31 so that, just as it is written, “LET HIM WHO BOASTS, BOAST IN THE LORD.”

Our ministry features a very helpful two-part article that addresses your concerns in much greater detail. We highly recommend you read it. You can find it here: Wrestling with God

Finally, we strongly encourage you to listen to the counsel of Scripture in our Romans study so you can fully grasp the meaning of predestination and the inevitable necessity of it given man's fallen nature and opposition to God. The doctrine is difficult, but it is also one of the most glorious truths of the Bible, because it amplifies the mercy and love of God knowing that He acted on our behalf while we were still His enemy:

Rom. 5:10 For if while we were enemies we were reconciled to God through the death of His Son, much more, having been reconciled, we shall be saved by His life.

Scripture quotations taken from the (NASB®) New American Standard Bible®, Copyright © 1995, 2020 by The Lockman Foundation. Used by permission. All rights reserved. www.lockman.org