Does John 3:1-5 teach that a person must be baptized in order to be saved?

Yes. That is correct for that time.

Joh 3:5 Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.

The advice Jesus gives Nicodemus does point to water baptism, and spirit baptism. There were conditions for salvation in the Kingdom gospel. In Acts, we see Peter preaching the same thing,

Act 2:38 Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost.

So, for salvation in the Kingdom gospel, it required faith in Jesus as Messiah and also to be water baptized and filled with the Holy Spirit. These were requirements to enter into the Millennial Kingdom when Jesus would come again in the Second Coming. The reason for these requirements is because of their significance for Israel. In Deut. and Lev., priests were washed to cleans them of sins before the ministered to God on behalf of the people. This is the origin of water baptism and is required for Jews as they will be the priests of God in the Millennial kingdom, Ex.19:5–6. Being filled with the Spirit is connected to the New Covenant when God will place a new heart of flesh into His people so that they would know His law and execute it over the nations in the Millennial kingdom, Ezel.36:24–27; Jer.31:31–34. Water and Spirit was significant and symbolic and identified the identity and the purpose of Jews in the coming kingdom.

However, please note that this is NOT the gospel for this dispensation of grace. We are not under the kingdom gospel. This gospel was suspended during the Acts period due to the unbelief of Israel. This is why God raised up Paul and sent him to the Gentiles. Today we (all men of all nations, incl. Israel) believe in Jesus Christ, placing our faith in the cross. It is by the death and resurrection of Jesus that we are saved. Simple faith in the cross work of Jesus is the only condition applicable in the grace gospel. We do not need to be water baptized or filled with the Holy Spirit because that is not part of our identity and purpose. The moment we get saved, we are baptized BY the Holy Spirit into the Body of Christ (1Co_12:13, no water needed!) Also, we are not filled with the Holy Spirit, but Christ Himself abides in us, Col_1:27.

Thus, it is important to recognize and understand the difference in what was taught in the gospel during Jesus' day and the gospel taught to us in Paul’s epistles.

I’ll summarize the gospels below,

Both the gospels require faith in Jesus Christ

• (Kingdom) One believed in the identity of Jesus as the Messiah and Son of God and one had to confess Him with the mouth. (Luke 12:8; Matthew 16:16; John 11:27)
  
  
• (Grace) One believes in the death, burial, and resurrection of Jesus Christ, 1Cor.15:1–4, that this sacrifice is the one and only atoning work of God to justify us and bring us to peace with God. (Rom 4:25; Rom 5:1–2)

Both the gospels required works

• (Kingdom) After faith one had to prove his faith with works. (Jam.2:19–26) Faith works included obedience to the Mosaic law, water baptism, filled with the Spirit, sell possessions, pick up cross and endure to the end.
  
  
• (Grace) The works required in this gospel is the already completed works of Jesus Christ. Jesus fulfilled the law on our behalf and His death and resurrection were payment for us. The works that Jesus did are imputed to us as if we did them, Rom.4:24; 2Cor.5:19. Jesus fulfilled the law as us. He died as us. He was raised as us. So, by imputation, we have fulfilled all the requirements that God needed and no more work for salvation is needed. In the grace gospel, we believe in Jesus and trust that His work is for our acquittal. What God expects of us when we get saved is to come to a knowledge of the truth (the Word, 1Tim.2:4). We are to be renewed in our mind (Rom.12:1–2) and allow the Word to fill us and influence our lives, Rom.8. Living out the Word which resides in us causes us to walk by faith or walk in the Spirit. God expects us as grace believers to grow up in faith and godliness as Christ is formed within us, Gal.4:19; Col.1:10.

I trust this provides some insights into the gospels and the importance of interpreting God’s Word according to how He dispensed it.