While we may all be bothered by an occasional doubt, it is a problem that must and can be overcome.
And the Bible tells us how to know whether we are saved.
There are several ways you can be absolutely sure that you are saved. First and foremost, you can know you're saved because Jesus Christ died for you. His blood cleanses all your sins. You can know you are saved because Jesus Christ, the Son of God, died to purchase your salvation. That is a historical fact. That is the saving work of Christ. If you trust in Christ, that salvation is yours.
You can also know you're saved because of the internal witness of the Holy Spirit. As it says in 1 John 5:6, "And it is the Spirit who bears witness because the Spirit is truth."
The Holy Spirit of God makes the truths about the work of Christ real to you. You see, God gave us the work of Christ. But to make the work of Christ real to us, He gave us the Spirit. Through faith, we receive the witness of God through the Spirit that Christ died for our sins and rose from the dead.
Finally, the Bible itself assures you that you're saved. Your salvation does not hinge on your emotions. You have an official record. You have the Word of God: "These things I have written to you who believe in the name of the Son of God, that you may know that you have eternal life, and that you may continue to believe in the name of the Son of God" (1 John 5:13).