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In the 1980's a certain bank with its headquarters in Edinburgh has a problem with (IBM) disc storage that had to be kept online for live customer account information for branch and ATM machine operation that it ran out of city buildings to put the disc drives in.

Yes: Not just a building, but buildings. Luckily, just after that time radical developments were made in disc storage densities and the need for more real estate diminished, but computer floor space was a big issue at the time.

Here, also, is a picture of the machine room at Manchester Computer Science, containing on single machine, the MU5. This is just the processor, the peripherals and disc storage are in another adjacent room:

^{_{Source: http://www.cs.manchester.ac.uk/about-us/history/mu5/}}

It was rather large, but the lower floor computer room that contained an ICL 1906A was even bigger; and then there was the CDC 7600 and the Cyber 106 too.

An earlier machine was Atlas. Here is a picture of the large room containing only the processor of the London University Atlas; several other large halls contained the peripherals and storage:

_{^{Source: http://www.chilton-computing.org.uk/acl/technology/atlas/p010.htm}}

They were all very big power hungry beasts that took some real estate.

There are plenty of examples on the internet in the computer history archives.

The main computer hall of the company I worked for in the 1970s and 1980s was about half the size of a soccer pitch - about 200 feet by 150 feet. That contained three IBM S/370 mainframes at one end, and the rest of the room was packed full of disk drives, stacked up to 6 or 7 feet high with narrow walkways between, with the outside walls lined with tape drives.

The power supplies and cooling systems filled the whole of the ground level, and the computer hall was the next floor up, built on a false floor to accommodate the wiring and plumbing for the water cooling.

To be fair, that was only half the complete building - one floor of the other half was an open plan area filled with punched card operators, and the other floor was occupied by programming teams.

The magnetic tape library occupied about half the machine hall area, on the third floor - basically, wall-to-wall racks of 12-inch tape reels, and a staff of tape librarians to make sure things didn't get lost!

At a later time there was also a Cray supercomputer in the main hall - though unless you knew where it was, it was almost hidden from view by all the rest of the kit.

But I can't seem to find any pictures of a computer filling an entire room, much less a whole multi-story building.

Well, for example look at this picture of a 4341 setup. This is a small entry-level mainframe of ~1980. I'd call that for sure a room full. The CPU itself is BTW the three half-height racks in the middle row.

Are these claims of a computer "filling an entire building" actually accurate, or is that a wild exaggeration?

As usual it depends on the size of building you look at. A fully configured mainframe of the 70s or 80s, with adequate peripherals, can easy fill 1000 m² (~11.000 sqft). Then again, companies using those kind of commercial computers usually had more than one machine.

Let's take a nice example of a mid to upper size bank system like I had a job with in 1981. They had a building the size of a Tesco Superstore (or one of these large DIY stores) to house 6 computers with all I/O and offices for machine operators and IT management. No user or any other department was located there. About 2/3 of that building was the machine room. 5 of them where used for daily business, while the 6th was a developer system. One of these five had the single job of operating a high speed optical reader, an awesome device ... anyway.

You see, they could get pretty big. A CPU (That's the mainframe term for the computer itself, processor, memory, memory interface, I/O controller and I/O interface - so without any peripheral device, not even a boot disk) did occupy four to six 21" full size racks, depending on the memory installed. In so far, these 1980s machines were already small, as the previous generation could have up to 10 racks just for the CPU. Later, around 1990, everything fitted in just 1-2 racks.

Consider ENIAC. From wikipedia:

It weighed more than 30 short tons (27 t), was roughly 2.4 m × 0.9 m ×
30 m (8 ft × 3 ft × 98 ft) in size, occupied 167 m2 (1,800 sq ft) and
consumed 150 kW of electricity.

That's roughly building-sized.

One of the largest computers ever built was the SAGE system, built to gather information about surprise attack on the US. It filled a building. One might argue that SAGE more than just one computer. But if you accept it as just one computer, it meets your criterion. Wikipedia Article

I can't remember where I saw or put it, but sometime in the last couple of months I came across an installation for an early IBM system somewhere online, in scanned PDF form (maybe in the Bitsavers documentation archive?), most likely their first "RAMAC" hard-drive using rig plus all the associated access and computing hardware for using it at the heart of an electronic accounting system.

It's some pretty heavy engineering. The first parts of it lay out the minimum allowable, and recommended dimensions for the room that the drive, the computer, the operator console, their various power supplies and glue logic racks will be installed into, in plan form, mapping out their positions (a multi-cabinet U-shape with the free-standing operator console nestled within) plus the floor-level cabling runs and the space needed for operators to get around the back of the machines to access service panels etc that would otherwise be sandwiched against the walls. I can't recall the minimum dimensions, but the recommended space wasn't far off the entire floor area of my apartment (which is about the same size as several previous workplaces, encompassing different offices and even medical examination rooms), pegging it around a square 25ft / 7-and-a-bit metres on each side. The access space around the "back" of the cabinets was no more than 4ft even so.

And that's for a relatively minimal system. Bear in mind that the term "mainframe" comes from an old term for the central processor of a computer - each of the system's primary components (CPU, memory, I/O handler, power regulator, etc) would be built into one or more of those large "frames", of which the processor (made up of a great many individual, interconnected rackmount cards stuffed full of discrete components or, much like HAL 9000, small-scale integrated circuit packages) was of course the "main" example. The singular encompassing the plural, it came to imply any complete computer system built along similar lines, regardless of size...

And of course as well as the plans there were photos and an artist's-impression birds-eye view of the machine in use. The two or three humans in the picture didn't look ever so big compared to the computer, but it was still sort of human-scale, instead of being a monstrous factory-filling item.

Of course, that's not literally filling the room, but it is taking up enough of it, in awkward enough positions, that you can't really make practical use of the space that's left unless you install the machine in the corner of a considerably larger room and employed the alternative L-shaped layout instead.

A "minicomputer" setup, by comparison, would have been one small enough to only occupy one or two such frames, or within a wider but lower piece of furniture analogous to a mainframe's operator console, with the processor, memory, offline storage devices, power supplies, and maybe even the user interface (where it wasn't a separate compact terminal or a teletype) all fitting within that limit.

Incidentally, you make reference to mainframes being "like" rackmount servers... actually the comparison is more direct than you might first imagine. The format of 19-inch communications and server (etc) rackmount frames is pretty much exactly the same as that of, particularly, IBM mainframes and/or DEC minicomputers. I can't remember if it's a direct heritage or just convergent evolution, but in either case it's the same environmental limitations that moulded them - to whit, they have to be able to fit through a standard doorway without getting stuck or damage being caused to equipment or building fabric, even if that may mean temporarily taking the door off its hinges. And it also needs to get around tight corners in corridors, stairwells, etc whilst in transit to the target room and being turned to fit through the door. But at the same time you want to have the maximum amount of useful space within each frame to minimise the floor area taken up by it, to maximise the amount of tech you can cram inside each box, and minimise the number of boxes and thus the building and shipping costs. So the format quite rapidly converges on the largest rectangular box that will fit through doorways and can be moved easily through buildings made to a common human scale. I think at one point it was even commonly called IBM Frame Size. But in general it's about 24 inches wide, a shade under 7ft high, and about 30 inches deep...

(I've even seen distinctly non-mainframey equipment clearly built to similar guidelines - like the back-projection mobile interactive whiteboards made by Smartboard back before short-throw lenses were good enough to make slimline front-projection models practical. They were quite a bit wider across the beam of course, but measured just less than a doorway's width front to back - a little less than a mainframe in fact, so you didn't have to take the doors off - were just short enough to fit under the lintel when lowered to their minimum height, and the castors and V-shaped integrated trolley compensated for the width by making it somewhat easier to "lever" around difficult corners and wiggle into rooms from narrow corridors... I expect all-up size was something similar to the largest part of the RAMAC setup, ie the massive, ~36-inch, horizontal-spindle hard disc unit proper, which needed a crane to lift on and off of trucks and planes and sometimes demanded remodelling of buildings to get it installed... like, holes would have to be made in walls and then bricked back up again afterwards, floors preemptively reinforced, etc. With the console being almost as bad, but at least being lower-rise and a sensible weight. Though the Smartboard was way lighter and moveable by one person, whereas even the pure logic parts of a mainframe usually needed a team of two or more...)