Does the Bible ever mention class systems?

There are no requirements in the New Testament for the establishment of or abolishment of socio-economic classes, so to the most basic aspect of your question, they can exist or not.

There are a few issues to address however. To begin, consider that elements of redistribution are seen directly in Acts:

32 Now the full number of those who believed were of one heart and soul, and no one said that any of the things that belonged to him was his own, but they had everything in common. 33 And with great power the apostles were giving their testimony to the resurrection of the Lord Jesus, and great grace was upon them all. 34 There was not a needy person among them, for as many as were owners of lands or houses sold them and brought the proceeds of what was sold 35 and laid it at the apostles' feet, and it was distributed to each as any had need. 36 Thus Joseph, who was also called by the apostles Barnabas (which means son of encouragement), a Levite, a native of Cyprus, 37 sold a field that belonged to him and brought the money and laid it at the apostles' feet. - Acts 4:32-37 ESV

This redistribution of goods comes with 2 notable caveats however:

1. It was entirely willful, and not a requirement placed on anyone.
2. It was limited to the body of believers.

In addition to giving within the church, Christians are also commanded by Christ to love our fellow man and to give freely to those in need, but again, it cannot be compulsory to be an act of faith. As Paul writes to the church in Corinth:

Each one must give as he has decided in his heart, not reluctantly or under compulsion, for God loves a cheerful giver. - 2 Corinthians 9:7 ESV

A classless society can exist within Christianity and it fact, it could be argued that no government is necessarily mutually exclusive with Christianity, as Christians are expected to submit to governing authority provided that such authority does not directly contradict what we are otherwise commanded by God to do:

1 Let every person be subject to the governing authorities. For there is no authority except from God, and those that exist have been instituted by God. 2 Therefore whoever resists the authorities resists what God has appointed, and those who resist will incur judgment. 3 For rulers are not a terror to good conduct, but to bad. Would you have no fear of the one who is in authority? Then do what is good, and you will receive his approval, 4 for he is God's servant for your good. But if you do wrong, be afraid, for he does not bear the sword in vain. For he is the servant of God, an avenger who carries out God's wrath on the wrongdoer. 5 Therefore one must be in subjection, not only to avoid God's wrath but also for the sake of conscience. 6 For because of this you also pay taxes, for the authorities are ministers of God, attending to this very thing. 7 Pay to all what is owed to them: taxes to whom taxes are owed, revenue to whom revenue is owed, respect to whom respect is owed, honor to whom honor is owed. - Romans 13:1-7 ESV

But remember:

27 And when they had brought them, they set them before the council. And the high priest questioned them, 28 saying, "We strictly charged you not to teach in this name, yet here you have filled Jerusalem with your teaching, and you intend to bring this man's blood upon us." 29 But Peter and the apostles answered, "We must obey God rather than men. - Acts 5:27-29 ESV

So, to summarize, Christianity exists independent of government and socio-economic structure and takes primacy to both, without placing restrictions on either.