Short answer
The Qur'an does not say so. It tells Noah to take on board "of each kind two, male and female". It gives no size for the ark and no count of the animals. Its word kull, "each" or "every", often means every thing of the kind in view. It says Noah was sent to his own people, and that those who rejected him were drowned. Most early scholars read the flood as covering the whole earth. Others read it as covering the land of Noah's people. The Qur'an's words do not settle it. So every species on earth packed into one boat is a reading, not something the Qur'an spells out.
Full explanation
What the Qur'an says
The Qur'an tells Noah's story many times. God (﷾) told him: "construct an Ark under Our eyes". When the flood came, He said: "Embark therein, of each kind two, male and female, and your family". The Qur'an gives no size for the ark. It gives no number of animals. It says Noah was sent "to his People" to warn them. It says God (﷾) drowned "those who rejected Our signs". When the water went down, the ark came to rest on Mount Judi.
“"But construct an Ark under Our eyes and Our inspiration, and address Me no (further) on behalf of those who are in sin: for they are about to be overwhelmed (in the Flood)."”
“At length, behold! there came Our command, and the fountains of the earth gushed forth! We said: "Embark therein, of each kind two, male and female, and your family - except those against whom the word has already gone forth,- and the Believers." but only a few believed with him.”
“We sent Noah to his People (with the Command): "Do thou warn thy People before there comes to them a grievous Penalty."”
“But they rejected him, and We delivered him, and those with him, in the Ark: but We overwhelmed in the flood those who rejected Our signs. They were indeed a blind people!”
“Then the word went forth: "O earth! swallow up thy water, and O sky! Withhold (thy rain)!" and the water abated, and the matter was ended. The Ark rested on Mount Judi, and the word went forth: "Away with those who do wrong!"”
Of each kind two, male and female
Noah was sent to his people
The size of the ark
The number of animals
Every species on earth
"Each" need not mean every species
The Arabic kull, "each" or "every", often means every thing of the kind in view. The same Arabic words come in 23:27, where Yusuf Ali writes "pairs of every species". The word there is again kull. The Qur'an says the Queen of Sheba was "provided with every requisite". Ibn Kathir explains: everything a powerful ruler needs. It says a wind would destroy "Everything", yet their houses still stood. So "of each kind two" does not by itself require every species on earth.
“So We inspired him (with this message): "Construct the Ark within Our sight and under Our guidance: then when comes Our Command, and the fountains of the earth gush forth, take thou on board pairs of every species, male and female, and thy family- except those of them against whom the Word has already gone forth: And address Me not in favour of the wrong-doers; for they shall be drowned (in the Flood).”
“At length, behold! there came Our command, and the fountains of the earth gushed forth! We said: "Embark therein, of each kind two, male and female, and your family - except those against whom the word has already gone forth,- and the Believers." but only a few believed with him.”
“"I found (there) a woman ruling over them and provided with every requisite; and she has a magnificent throne.”
Explains that the Queen of Sheba was given "all things" as: all the conveniences of this world that a powerful monarch could need.
“"Everything will it destroy by the command of its Lord!" Then by the morning they - nothing was to be seen but (the ruins of) their houses! thus do We recompense those given to sin!”
Explains the wind "destroying every thing" as destroying every part of their land that could normally be destroyed; afterwards nothing was seen but their dwellings.
How the scholars have read the flood
Ibn Kathir, the fourteenth-century commentator, says Noah was told to take a pair of every kind of creature with a soul. Some, he reports, added plants. Ibn Atiyya, a commentator of twelfth-century Spain, reports two views. One group said the flood drowned only part of the earth, the land of Noah's people. The majority said it drowned the whole inhabited earth. Ibn Atiyya thought this the plainer meaning. So Muslim scholars have held both views. The Qur'an itself does not settle it.
Allah commanded Noah to take on the ship one pair of every kind of creature possessing a soul. Some said that this included other creatures as well, such as pairs of plants, male and female.
Reports two views of the flood. One group said the flood drowned only a part of the earth, the part belonging to Noah's people. Another group, the majority, said it drowned the whole inhabited earth. Ibn Atiyya judges this second view the apparent meaning, because Noah built the ship and sent out a bird to see whether the water had gone.
Noah's prayer, and the land
Two verses are often read as a flood over the whole world. Noah prayed: "Leave not of the Unbelievers, a single one on earth!" (71:26). And God (﷾) says He "made his progeny to endure (on this earth)" (37:77). There the words "on this earth" are Yusuf Ali's own, in brackets. The Arabic word for earth in 71:26, al-ard, can also mean the land in view. Pharaoh's chiefs say Moses plans "to get you out of your land" (7:110). Pharaoh "elated himself in the land" (28:4). Both use the same word, ard. So Noah's prayer may speak of the land of his own people. Most early scholars still held that the flood covered the whole earth. The Qur'an's words allow both readings.
“And Noah, said: "O my Lord! Leave not of the Unbelievers, a single one on earth!”
“And made his progeny to endure (on this earth);”
“"His plan is to get you out of your land: then what is it ye counsel?"”
“Truly Pharaoh elated himself in the land and broke up its people into sections, depressing a small group among them: their sons he slew, but he kept alive their females: for he was indeed a maker of mischief.”
Reports two views of the flood. One group said the flood drowned only a part of the earth, the part belonging to Noah's people. Another group, the majority, said it drowned the whole inhabited earth. Ibn Atiyya judges this second view the apparent meaning, because Noah built the ship and sent out a bird to see whether the water had gone.
Perspectives
The critics' reading
Critics say one boat could not hold a pair of every species on earth, so the story cannot be true.
No boat could carry a pair of every species on earth.
Rebuttal
The Qur'an gives no size and no count, and "of each kind" need not mean every species on earth. Some scholars read the flood as covering the land of Noah's people only.
The Islamic perspective
The Qur'an says Noah took on board "of each kind two, male and female". It gives no size and no count, and its word for "each" need not mean every species on earth. Scholars have read the flood as worldwide and as local.
The Qur'an tells Noah to take "of each kind two, male and female", and gives no size for the ark and no number of animals.
Kull often means every thing of the kind in view, not every single thing.
Classical scholars differed: the majority read the flood as covering the inhabited earth, a group read it as covering the land of Noah's people.