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If you read about the history of computing, you'll hear how the first computers were "huge". You will often come across assertions that in the early days of commercial computing, a single computer would be "so big that it filled an entire building".

Now, poking around Wikipedia, I can find plenty of photos of old computers the size of an entire server rack, or several server racks. But I can't seem to find any pictures of a computer filling an entire room, much less a whole multi-story building.

Are these claims of a computer "filling an entire building" actually accurate, or is that a wild exaggeration? I can well imagine if you just paid a few million USD for a computer, you probably put it in its own special room with locked doors. But do any of these systems really fill a whole building? Do any of them really "fill" a whole room? Most pictures seem to just show a mostly empty room with cabinets across one wall.

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