The Big Bad is a Computer destroying Techwhiz. The Good Guy's on the other side. Now, how do you show the audience that they are awesome, other than having them grab a keyboard, punch a few buttons, and blow up a satellitein 15 minutes?
Throw in the 21st-century equivalent of the Mad Scientist Laboratory, and have them live in a room packed to the brim full of technological things, that's what. A Hacker Cave, known in chanspeak as a 'battlestation'
Note that said definition of "awesome" must include one of the following words in its description (amongst others): Geeky, Technology, Things that go BEEP, Metallic, futuristic, pipes, blinkenlights, Billions of Buttons. Anime figurines and other otaku paraphernalia does apply. Must include more than one obvious computer tower (because the hacker knows computers don'tequal monitors), and the primary computer comes with at least 3 monitors (never mind some of those monitors display only a looping picture that does nothing). May also include a retro coin-op arcade game (Pac-Man and Space Invaders are obvious favorites) in a corner, just because.
When done well, the result will be unabashed Technology Porn.
"Real" hackers spend most of their hacking time peering at a text editor and thinking very hard, so one might think this unrealistic, but it still manages to be Truth in Television for many. A common cause is the need to keep any number of test machines on hand to verify programs' behavior on dissimilar platforms, or in complex networked setups. Demoparties and other hackish social occasions often resemble this trope too- as do the bedrooms (or basements) of hackers who deliberately seek this trope as an ideal!
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Anime & Manga
Noah & Seto Kaiba from Yu-Gi-Oh! both have one in the anime.
Lain's set-up in Serial Experiments Lain. Now, how did Lain, a 14-year-old schoolgirl, got all those $2000+ Cisco Catalyst switches running the latest IOS?
They grew there out of nothing. Her computer has grown so much in later episodes that it's started to sprout from the outside walls like a plant.
Given that Lain's parents are fake parents meant to "incubate" her, they can probably afford to spend money on that kind of stuff because it's their purpose.
Satsuki in X1999 has only one giant computer, but it's still in a cooled room under the Tokyo City Hall.
The control room of L's high-rise, though the large monitors are generally used for observing the security cameras of the building. There are only three actual computers with one monitor each, which makes perfect sense when you consider there are usually several people working at once.
Chisame of Mahou Sensei Negima!'s Magitek pactio item can create a perfect hacking space for her inside a Tron-like area complete with surrounding screens, numerous floating keybords and seven electronic helpers. Her own room surrounded by monitors probably also counts.
In the series she visits a hacker, who lives in the more typical dark, cramped room. This particular Hacker Cave is filled with (deliberately) anachronistic computer equipment, a good proportion of which is obsolete. There were also at least two robotic sex-dolls lying around.
The Digimon Kaiser's headquarters in Digimon Adventure 02 is a rather strange instance of this trope - it's basically this, but the screens are kind of just floating there in featureless blackness and all the controls are on the arms of the Kaiser's chair. The room is apparently pretty small but otherwise has no distinguishing features. Its strangeness may be justified by it being in the Digital World instead of the real world.
Yamaki of Digimon Tamers is a black ops agent and computer programmer. His Hacker Cave is the HYPNOS headquarters, whose operation's center filled to the brim with computers, wires, monitors, and even a massive overhead screen. In the beginning of the series, Yamaki is so obsessed with Digimon, he seems to never leave the place. When we finally see him at his apartment being fired but overcoming his obsession, he sets up a computer and accessories in his living room.
Oracle, from DC Comics, builds one of these for herself wherever she goes. The most famous of which was the Clocktower in Gotham City, original headquarters of the Birds of Prey.
The Beetlecave! Jaime's hacker buddies, Hector and Nadia, basically turned their entire house into this. The actual 'Beetlecave', however, is the cyberspace face of it. All because Jaime's parents probably wouldn't like their son constructing a real cave in the crawlspace...
Die Hard 4.0 had a couple of these, all the hackers that blow up at the start (at least the ones we see) imply they have a mass of pimped-out computers, then there is the truck the bad guys go around in. And again in the Woodlawn building, it has a massive super server to hold all the financial transactions of the USA (yet is able to be downloaded to a single laptop).
Cryptonomicon arguably sees Lawrence Pritchard Waterhouse creating the world's first hacker cave when in the later stages of World War II his superiors discover his lab to be taken over by a massive and revolutionary self-built computer through which he has been running literal truckloads of punch cards to aid in his codebreaking work.
The Evil Genius Trilogy has a couple, With Hardware Heaven in the first book, and The War Room in the second.
Live Action TV
The IT department in The IT Crowd is full of obsolete (but notorious) computer equipment, cabling, geeky posters and the various bits of debris that a computer room accumulates.
The Watchtower in the last few seasons of Smallville was this for Chloe. Back in the high school days of the early seasons, the Smallville High Torch office was this for her as well, though in that case it was open to all students and faculty rather than being private property (it really was more a matter of Chloe being the only one who consistently wanted to be at the Torch!)
Lana also got one in Season 7, as she used it to spy on Lex's every move.
The Shadow Broker has a pretty impressive one in Mass Effect 2: Lair of the Shadow Broker. Liara manages to keep her hands on most of the hardware, setting it up in Miranda's old office by the time of Mass Effect 3.
In Deus Ex: Human Revolution, Dutch hacker Arie van Bruggen has two. His first setup is in his penthouse apartment hidden behind a false wall. His second, after he's forced to abandon the first one and go into hiding, is in the Alice Garden Pods capsule hotel, which is impressive in that one "room" in that place is nothing more than a bed, a closet the size of a cupboard, a bookshelf, and a curtain for privacy. Van Bruggen got around this by renting every single pod adjacent to his, then packing them completely full with server towers.
Wade's room in Kim Possible, which is usually only seen from the POV of his webcam.
Code Lyoko: One of the few Hacker Caves that is actually underground, apparently.
And hollowed out from an abandoned auto assembly plant.
Lampshaded slightly in Invader Zim in the episode "Zim eats waffles". Dib is shown in his own hacker cave trying to record Zim's evil plans. Only to have his various hard drives miss the crucial moments of video.
Server rooms can get to look like this in real life; they tend to be located in the basement or some other room without much natural light, and usually double as storage for spare workstations and replacement parts and assorted mysterious bits and pieces that nobody ever gets around to getting rid of. Sysadmins exiled down there because they're considered unworthy of a proper office in more congenial surroundings sometimes brighten the place up a bit with posters and memorabilia, further enhancing the resemblance to this trope.
Some graphic designers/artists utilize this kind of setup too, as this photo shows us.