ANSWER: The short answer is that the Bible has NOTHING to say about Hell and Purgatory (as they are traditionally taught) because these doctrines did not come from its pages. They came primarily from the Roman Catholic Church. They are built upon assumptions, traditions, and misunderstandings that are ultimately void of sound Scriptural support.
Before we tackle the doctrine of purgatory let us discover the Biblical TRUTH regarding Hell. In the Old Testament, the original language word translated for this location is Sheol. It's literal meaning (Strong's Concordance #H7585) is that it is a reference to a common grave. The word has no relationship whatsoever to our modern definition of a place of eternal punishing and punishment. This definition of hell as a place of eternal torment comes to us influenced greatly by a book named the Divine Comedy, written by Dante, around 1300 A.D.
In the New Testament, only three Greek words are translated as HELL. The first is Tartaroo, Strong's Concordance #G5020, which is used only once in the New Testament, in 2Peter 2:4, and means the deepest part of Hades. The second is Hades, Strong's Concordance #G86, which is exactly equivalent to the Hebrew word Sheol and has an identical meaning (See Acts 2:27, Revelation 1:18, 20:13, and other verses). The last is Ghenna (Gehenna), Strong's Concordance #G1067, which refers to the Valley of Hinnom located outside the gates of Jerusalem. Jesus referred to this place in Matthew 5:22, 18:9 and other verses.
Gehenna, translated as Hell in the KJV translation, was the name of a deep, narrow valley that existed on Jerusalem's southern end. In ancient times Israelites, and even some kings of Israel, used the valley to practice their idolatry to pagan gods such was Baal and Molech. One way these evil "gods" were worshipped and appeased was through the completely Satanic practice of burning CHILDREN, include little children and babies, alive by placing them in the red hot hands of these idols (for more information read 2Kings 16, 23, and 2Chronicles 28).
The valley eventually was held in such disgust that Jews returning from captivity would use it to get rid of their trash. Dead bodies and carcasses were also flung into the ravine. The valley of Hinnom was also a place where public executions were held. By the time of Jesus, the area had become incredibly offensive. Fires were kept burning in the valley day and night (the UNQUENCHABLE FIRE Jesus referred to in Matthew 3:12 and Luke 3:17) in order to consume what was put in it. Jesus was referring to the Hinnom valley when he warned against the punishment of Hell to those who put down others (Matthew 5:22).
The Bible teaches that after the Millennium those who are incorrigibly evil and who deliberately turn away from God (they commit the unpardonable sin) will be destroyed by fire in a punishment that will be eternal. This is a ONE TIME punishment that cannot be undone.
An intermediary state?
In regard to Purgatory, Catholics teach that some who sin and repent may still be owed, at death, earthy punishments for their transgressions (even though God forgave them). The "souls" of these individuals, according to the church, while not worthy of eternal torment, are not yet "clean" enough to enter heaven and present themselves before the Eternal. Their solution to this problem is that such a person must enter an intermediary state of existence, known as purgatory, in order to be purged from all sin and made spiritually clean. It is only after this purging, according to the Catholics, that a person is worthy enough to enter heaven.
The church defends their teaching by arguing that prayer for the dead is an ancient practice (actually, it is an ancient FALSE practice promulgated by those considered 'authorities' by the church). Its practice assumes some of those who have died are in a state of suffering that prayer by the living can somehow lessen.
Scripture, however, does not even mention the existence of this place or that even the prayers of the living can somehow BENEFIT those who have died (who are NOT in heaven but still in their graves awaiting a resurrection). Additionally, the teaching of a purgatory makes a mockery of Jesus' sacrifice! His sinless sacrifice as the Lamb of God makes possible the COMPLETE and TOTAL cleansing of ALL sins for those who repent (1John 1:7, 9, etc.).
In conclusion, the belief in an eternal hell of torture and torment, or a "middle state" of purgatory, cannot be supported in the Bible. These are the deceived teachings of men and not the inspired doctrines of God!