What do 1st century extra-biblical sources say about hell?

Early Christians:

Epistle of Barnabas (70 - 130 AD):

The way of darkness is crooked, and it is full of cursing. It is the way of eternal death with punishment. (“Epistle of Barnabas”)

Ignatius of Antioch (110 AD):

Corrupters of families will not inherit the kingdom of God. And if they who do these things according to the flesh suffer death. how much more if a man corrupt by evil reaching the faith of God. for the sake of which Jesus Christ was crucified? A man become so foul will depart into unquenchable fire: and so will anyone who listens to him. (Letter to the Ephesians 16:1-2)

Clement of Rome (150 AD):

If we do the will of Christ, we shall obtain rest; but if not, if we neglect his commandments, nothing will rescue us from eternal punishment (“Second Clement” 5:5)

But when they see how those who have sinned and who have denied Jesus by their words or by their deeds are punished with terrible torture in unquenchable fire, the righteous, who have done good, and who have endured tortures and have hated the luxuries of life, will give glory to their God saying, 'There shall be hope for him that has served God with all his heart!' (“Second Clement” 17:7)

The Martyrdom of Polycarp (155 AD):

Fixing their minds on the grace of Christ, [the martyrs] despised worldly tortures and purchased eternal life with but a single hour. To them, the fire of their cruel torturers was cold. They kept before their eyes their escape from the eternal and unquenchable fire (“Martyrdom of Polycarp” 2:3)

Irenaeus (189 AD):

Christ Jesus, our Lord, and God, and Saviour, and King, according to the will of the invisible Father, 'every knee should bow, of things in heaven,, and things in earth, and things under the earth, and that every tongue should confess' to Him, and that He should execute just judgment towards all; that He may send 'spiritual wickednesses,' and the angels who transgressed and became apostates, together with the ungodly, and unrighteous, and wicked, and profane among men, into everlasting fire; but may, in the exercise of His grace, confer immortality on the righteous, and holy, and those who have kept His commandments, and have persevered in His love, some from the beginning of their Christian course, and others from the date of their repentance, and may surround them with everlasting glory. (“Against Heresies” 1:10:10)

The penalty increases for those who do not believe the Word of God and despise his coming. . . . [I]t is not merely temporal, but eternal. To whomsoever the Lord shall say, ‘Depart from me, accursed ones, into the everlasting fire,’ they will be damned forever (“Against Heresies” 4:28:2)

Tertullian (197 AD):

These have further set before us the proofs He has given of His majesty in judgments by floods and fires, the rules appointed by Him for securing His favor, as well as the retribution in store for the ignoring, forsaking and keeping them, as being about at the end of all to adjudge His worshippers to everlasting life, and the wicked to the doom of fire at once without ending and without break, raising up again all the dead from the beginning, reforming and renewing them with the object of awarding either recompense. (“Apology” 18:3)

Then will the entire race of men be restored to receive its just deserts according to what it has merited in this period of good and evil, and thereafter to have these paid out in an immeasurable and unending eternity. Then there will be neither death again nor resurrection again, but we shall be always the same as we are now, without changing. The worshipers of God shall always be with God, clothed in the proper substance of eternity. But the godless and those who have not turned wholly to God will be punished in fire equally unending, and they shall have from the very nature of this fire, divine as it were, a supply of incorruptibility (“Apology” 44:12–13)

Therefore after this there is neither death nor repeated resurrections, but we shall be the same that we are now, and still unchanged--the servants of God, ever with God, clothed upon with the proper substance of eternity; but the profane, and all who are not true worshippers of God, in like manner shall be consigned to the punishment of everlasting fire--that fire which, from its very nature indeed, directly ministers to their incorruptibility. ("Apology" 48:12)

There are many more but I think these should help you.

As for Jewish sources, I am still looking for sources. Once I do, I will post them here.

I am still looking for good material from the 1st century about hell from Jewish sources. It just so happens that the concept of Heaven and Hell is a more Christian concept than it ever was Jewish and rabbis have had very little to say, if anything at all, over the vast history of Judaism (which is still a lot but little compared to Christianity). Anyway, I found this page that you will find helpful, I think:

Heaven and Hell in Jewish Tradition.

Excerpt from the above link:

There are, however, several biblical references to a place called Sheol (cf. Numbers 30, 33). It is described as a region "dark and deep," "the Pit," and "the Land of Forgetfulness," where human beings descend after death. The suggestion is that in the netherworld of Sheol, the deceased, although cut off from God and humankind, live on in some shadowy state of existence. While this vision of Sheol is rather bleak (setting precedents for later Jewish and Christian ideas of an underground hell) there is generally no concept of judgment or reward and punishment attached to it. In fact, the more pessimistic books of the Bible, such as Ecclesiastes and Job, insist that all of the dead go down to Sheol, whether good or evil, rich or poor, slave or free man (Job 3:11-19).

And then:

The development of the concept of life after death is related to the development of eschatology (speculation about the "end of days") in Judaism. Beginning in the period following the destruction of the First Temple in Jerusalem (586 BCE), several of the classical Israelite prophets (Amos, Hosea, and Isaiah) began forecasting a better future for their people. However, with repeated military defeats and episodes of exile and dislocation culminating in the destruction of the Second Temple in 70 CE, Jewish thinkers began to lose hope in any immediate change, instead investing greater expectations in a messianic future and in life after death. This was coupled with the introduction into Judaism of Hellenistic notions of the division of the material, perishable body and the spiritual, eternal soul. The catastrophe of 70 CE caused a theological crisis. How could it be that the God of Israel would simply allow His sanctuary to be destroyed and His people to be vanquished at the hands of the Roman Empire? While the rabbis often claimed that it was the Israelites' sinfulness that led God to allow it to be defeated (mi-p'nei hataeinu, "because of our sins"), it was more difficult to explain why good and decent individual Jews were made to suffer.

It talks about Jewish "tradition" but that's because the Tanakh does not address heaven and hell, as we understand it, in much detail.

The above is not in any way "the" Jewish belief. Its just an article giving you insight into some of the thought and discussions of the Talmud and the rabbis.

If and when I find direct quotes, I will put them here. But I really hope this helps you in your study/research.