Is Heaven up and Hell down?

Given the flat-earth conception of the universe that has already been mentioned, heaven is "up" in the "heavens" and hell/Sheol is "down" beneath the ground. The argument for "not knowing exactly where heaven is" is a modern problem that has only existed since the time of Galileo. Before the age of the telescope, heaven - as in the place where God lived - was widely accepted to be the sky and the clouds. This fact is clearly demonstrated in several places in the Bible, some of which have been cited in the other answers.

In short, the only view of heaven that is consistent with the Bible's actual teachings is "the sky". Yahweh is, quite literally, a sky god.

An Analysis of the Word Heaven

In the beginning God created the heavens and the earth. (Genesis 1:1 NKJV)

The Hebrew word for Heaven is (הַשָּׁמַ֖יִם) meaning: "Heaven, Sky"

So since the "Visible Sky" can be "Up", heaven "Can be up". Now if you are upside down, that's a different story ^^

An Analysis of the Word Hell

The Hebrew word for Hell is (שְׁאוֹל) "Sheol" meaning "Grave, Dirt-Pit"

The Greek word for Hell (ᾍδης) "Hades" meaning "Grave, Dirt-Pit"

The Greek word for Hell (γέεννα) "Gehenna" a place where bodies where Cremated

So since the "Grave" can be "down", Hell "Can be down".

Why this idea of Eternal Punishment?

One time the Greek Word (ταρταρώσας) "Tartarus" was translated as Hell and considered a place of punishment for the dead.

This verse translates Tartarus as Hell

For if God did not spare the angels who sinned, but cast them down to hell and delivered them into chains of darkness, to be reserved for judgment; (2 Peter 2:4 NKJV)

When you group the concepts of the Grave with a crematory with Eternal Punishment
You end up with today's understanding of Hell.

Here we can find a clearly defined example of the punishment

For the wages of sin is death, but the gift of God is eternal life in Christ Jesus our Lord. (Romans 6:23 NKJV)

but of the tree of the knowledge of good and evil you shall not eat, for in the day that you eat of it you shall surely die. (Genesis 2:17 NKJV)

Jesus said to her, “I am the resurrection and the life. He who believes in Me, though he may die, he shall live. 26 And whoever lives and believes in Me shall never die. Do you believe this?” (John 11:25-26 NKJV)

For he who sows to his flesh will of the flesh reap corruption, but he who sows to the Spirit will of the Spirit reap everlasting life. (Galatians 6:8 NKJV)

The Punishment of Death

Opening

In the Apostles' Creed, in the fifth article of the creed, [Jesus Christ] descended into hell, and in next article, the sixth, he ascended into heaven.

Thus from our deposit of faith, heaven is where people ascend [up] into and hell where people descend [down] into . But as will be answered below, the Church has decided nothing as regards their exact locale.

Where is heaven, the dwelling of God and the blessed?

Please see The location of Heaven in Heaven | New Advent.

From the article:

In general, however, theologians deem more appropriate that there should be a special and glorious abode, in which the blessed have their peculiar home and where they usually abide, even though they be free to go about in this world. For the surroundings in the midst of which the blessed have their dwelling must be in accordance with their happy state; and the internal union of charity which joins them in affection must find its outward expression in community of habitation. At the end of the world, the earth together with the celestial bodies will be gloriously transformed into a part of the dwelling-place of the blessed (Revelation 21). Hence there seems to be no sufficient reason for attributing a metaphorical sense to those numerous utterances of the Bible which suggest a definite dwelling-place of the blessed. Theologians, therefore, generally hold that the heaven of the blessed is a special place with definite limits. Naturally, this place is held to exist, not within the earth, but, in accordance with the expressions of Scripture, without and beyond its limits. All further details regarding its locality are quite uncertain. The Church has decided nothing on this subject.

Where is hell?

Please see the paragraph that starts - Where is hell? - in the section Name and place of hell in Hell | New Advent.

Short answer, no one knows.

From the article:

The Bible seems to indicate that hell is within the earth, for it describes hell as an abyss to which the wicked descend. We even read of the earth opening and of the wicked sinking down into hell (Numbers 16:31 sqq.; Psalm 55:15; Isaiah 5:14; Ezekiel 26:20; Philippians 2:10, etc.). Is this merely a metaphor to illustrate the state of separation from God? Although God is omnipresent, He is said to dwell in heaven, because the light and grandeur of the stars and the firmament are the brightest manifestations of His infinite splendour. But the damned are utterly estranged from God; hence their abode is said to be as remote as possible from his dwelling, far from heaven above and its light, and consequently hidden away in the dark abysses of the earth. However, no cogent reason has been advanced for accepting a metaphorical interpretation in preference to the most natural meaning of the words of Scripture. Hence theologians generally accept the opinion that hell is really within the earth. The Church has decided nothing on this subject; hence we may say hell is a definite place; but where it is, we do not know.

And this is my opinion as well.