Here is a partial answer with the evidence I found so far.
I remember reading in the novel someone asked how large the Star Cluster was and they were told it was 1,500 feet (457.2 meters) long. They wondered how much that was exaggerated. And the narrator said the true size was 10 times that much. If the ship was 10 times that long, it would be 15,000 feet (4,572 meters). If the length of the ship was increased by 10 times, and the proportions remained the same, the volume would increase by 1,000 times. The cube root of 10 is about 2.1544, and 2.1544 times 1,500 feet is 3,231.6 feet (984.99168 meters).
"Concealment", "The Storm", and "The Mixed Men" are available at:
http://baencd.freedoors.org/Books/Transgalactic/1416520899___9.htm
http://baencd.freedoors.org/Books/Transgalactic/1416520899__10.htm
http://baencd.freedoors.org/Books/Transgalactic/1416520899__11.htm
"Concealment" has a paragraph:
She saw that the men were leaving the meteorite. Decisively, she clicked off the intership communicator, made an adjustment and stepped through a transmitter into the receiving room half a mile distant.
Half a mile is 2,640 feet (804.672 meters), so it is possible to travel at least that far within the ship.
The receiving room is large:
She watched the building take shape inside the large receiver. It slid out finally on wings of antigravity, and was deposited in the center of the enormous metal floor.
The senior ship meteorologist, Lieutenant Cannons, stood up from a chair as she came toward him across the vast floor of the transmission receiving room, where the Fifty Suns weather station still stood.
In "The Storm":
They had stopped looking at the map. Maltby stood with the councilors at the great window staring up at the Earth ship.
The machine was scarcely more than a dark sliver in the distant sky. But the sight of it seemed to hold a deadly fascination for the older men.
Maltby sees the Star Cluster from a closer distance:
It was directly above him now that he had come so many miles across the city toward it. It was tremendously high up, a long, black torpedo shape almost lost in the mist of distance.
Both those descriptions imply that the length is several times the width.
The meterorology room aboard is large:
Because abruptly he was in a very large room.
As the Star Cluster approaches the storm:
In those minutes before disaster struck, the battleship Star Cluster glowed like an immense and brilliant jewel. The warning glare from the Nova set off an incredible roar of emergency clamor through all of her hundred and twenty decks.
From end to end her lights flicked on. They burned row by row straight across her four thousand feet of length with the hard tinkle of cut gems. In the reflection of that light, the black mountain that was her hull looked like the fabulous planet of Cassidor, her destination, as seen at night from a far darkness, sown with diamond shining cities.
If the 120 decks are 10 to 15 feet (3.048 to 4.572 meters) from deck to deck, The ship would be 1,200 to 1,800 feet (365.76 to 548.64 meters) tall. If it had a circular cross section it would be 1,130,000 to 2,540,000 square feet (105,000 to 236,000 square meters). If it had a square cross section it would be 1,440,000 to 3,240,000 square feet (133,480.37 to 301,005.84 square meters).
If the Star Cluster is 4,000 feet (1,219.2 meters) long, and has a circular cross section it should have a volume of 4,520,000,000 to 10,160,000,000 cubic feet (128,016,000 to 287,731,200 cubic meters). If the Star Cluster is 4,000 feet (1219.2 meters) long, and has a square cross section, it should have a volume of 5,760,000,000 to 12,960,000,000 cubic feet (162,739,669.4 to 366,986,320.1 cubic meters). Assuming that it has blunt ends and does not taper, of course.
If the Star Cluster is 4,000 feet (1,219.2 meters) long, and 1,200 to 1,800 feet (365.76 to 548.64 meters) tall, it should be about 2.2222 to 3.3333 times as long as it is tall. And that doesn't seem very narrow to me. Maybe the Star Cluster is several times as tall as it is wide, so it looks skinner when seen from below.
As the Star Cluster slams into the storm:
She began to break up.
And still everything was according to the original purpose of the superb engineering firm that had built her. The limit of unit strain reached, she dissolved into her nine thousand separate sections. Streamlined needles of metal were those sections, four hundred feet long, forty feet wide; sliverlike shapes that sinuated cunningly through the gases, letting the pressure of them slide off their smooth hides.
So the steamlined sections were 400 feet (121.92 meters) long by 40 feet (12.192 meters) wide.
If each section had a circular cross section, it would be 1,256.64 square feet (116.52 square meters) and the volume would be 502,656 cubic feet (14,206.118 cubic meters). If each section had a square cross section, it would be 1,600 square feet (148.64486 square meters) and the volume would be 640,000 cubic feet (18,122.781 cubic meters). Assuming that it has blunt ends and does not taper, of course.
So 9,000 of such sections of the ship would have a total volume of 4,523,904,000 to 5,760,000,000 cubic feet (127,855,062 to 163,105,029 cubic meters) if they had circular or square cross sections.
Above it was calculated that if the Star Cluster is 4,000 feet (1219.2 meters) long, and has a square cross section, it should have a volume of 5,760,000,000 to 12,960,000,000 cubic feet (162,739,669.4 to 366,986,320.1 cubic meters). And just now it is calculated that if the 9,000 sections each had a square cross section the total folume would be 5,760,000 cubic.
If the Star Cluster is 4,000 feet (1219.2 meters) long, it would have a length of 10 of the sections it could divide into. Therefore it would have to have an average cross section of 900 of those sections, which would be 30 by 30 sections, or 1,200 by 1,200 feet (365.76 by 365.76 meters).
If the Star Cluster is 120 decks tall, and each deck is 10 feet tall, 4 decks coudl fit into each section, and the ship would be 30 sections tall. That also adds up.
But I can't help wondering whether the ship was actually longer and narrower. If it was 10 secitons wide by 10 sections high, it would have 100 sections in its cross section and would have to be 90 sections long, or 36,000 feet (10,972.8 meters).
If it was 10 sections wide by 9 sections high (400 by 360 feeet) it would have 90 sections in its ross section and be 100 sections long, or 40,000 feet (12,192 meters). And possibly 4,000 feet was a mistake for 40,000 feet. But that would make it only 36 decks high.
If it was 22 sections wide by 4 sections high, and each section was 3 decks high, there would be 12 decks, and 120 decks could be a typo for 12 decks. The cross section would 88 sections, so if it was 100 sections and 40,000 feet long there would be a total of 8,800 sections.
If each section was 1 deck high, the ship could have a cross section of 8 by 12 sections and be 320 feet wide by 480 feet high with 12 decks. It would have 96 sections in its cross section and thus have a total of 9,024 sections if it was 94 sections, or 37,600 feet (11,460 meters) long.
I note that if the Star Cluster was 3,231.6 feet (984.99168 meters) long, a possible length calculated above, that would 8.079 section lengths long. If the ship was 8 sections or 32,000 feet long, it would be even fatter than the fattest shape calculated above.
If the ship was 15,000 feet (4,572 meters) long, it would be 37.5 section lengths long. 8,880 sections divided by 37 gives a cross sectiono of 240 sections or 15 by 16 sections, and 8,550 sections divided by 38 sections gives a cross section of 225 sections, or 15 by 15 sections.
So there are some problems with the dimensions of the Star Cluster, and I would appreciate any other data from the stories or novel about its dimensions.
