The Bible itself answers the question:
Who are the elect?

With the help of an Interlinear Bible*, the Greek word can be found that is translated as "elect". With an Analytical Concordance we can find the places where else in the Bible that specific word is used.

eklektos Greek Strong's NR. 1588; select, by impl. favorite:— choice(2), choice man(1), chosen(1), chosen(9), chosen one(1), elect(8).

In the whole NT this word is used in 22 places: (Matthew 22:14; Matthew 24:22,24,31; Mark 13:20,22,27; Luke 18:7, 23:25; Romans 8:33; 16:13; Colossians 3:12; 1 Timothy 5:21; 2 Timothy 2:10; Titus 1:1; 1 Peter 1:1, 2:4,6,9; 2 John 1:1,13 and in Revelation 17:14)

Reading all the verses that use the Greek word eklektos brings out the answer. Romans 8 summarizes it in a clear way:

Romans 8:15 For you have not received a spirit of slavery leading to fear again, but you have received a spirit of adoption as sons by which we cry out, "Abba! Father!" In his letter to the Romans Paul explains that God has set us free and adopted us as sons (Jews and Gentiles who have put their believe in Jesus) and are allowed to cry out "Abba! Father!" He goes on to say that [16] The Spirit Himself testifies with our spirit that we are children of God, and in [21] that the creation itself also will be set free from its slavery to corruption into the freedom of the glory of the children of God.
The next few verses identify exactly who the scripture points to when it talks about the elect:
[29] For those whom He foreknew, He also predestined to become conformed to the image of His Son, so that He would be the firstborn among many brethren; [30] and these whom He predestined, He also called; and these whom He called, He also justified; and these whom He justified, He also glorified.
[33] Who will bring a charge against God's elect? God is the one who justifies;

About whom is verse 29 speaking? The answer is in Paul's letter to the Ephesians: God choose us in HIM and adopted us as sons through Jesus Christ.
Ephes. 1:3-6 Blessed be the God and Father of our Lord Jesus Christ, who has blessed us with every spiritual blessing in the heavenly places in Christ, [4] just as He chose us in Him before the foundation of the world, that we would be holy and blameless before Him. In love [5] He predestined us to adoption as sons through Jesus Christ to Himself, according to the kind intention of His will, [6] to the praise of the glory of His grace, which He freely bestowed on us in the Beloved.

The elect are all those who through their trust in Jesus Christ have become children of God.

How do you become part of the elect?

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*
a Bible that contains, word for word, both the original Greek text and its English translation but does not bother with English grammar.