Clarification : Due to genre conventions (the Noir genre) - there aren’t costumes. The essential set-up is that there’s an elite privately-funded investigation agency that specializes in the paranormal - whether that be magical, or superhuman. The PCs will all be new recruits to this organization.
If that helps anybody. 
What’s the certain NYC block?
If this hasn’t already been concretely decided, may I suggest Chinatown in Manhattan? It’s one of the few areas least gentrified, it has a wonderful ethnic flavor that can suggest a noir atmsphere almost immediately appropiate for both the present and any flashback sequences, the people living there are notoriously uncooperative with the police, and you can populate the area above ground with one or two ethnic characters privvy to the secrets of the base but deflecting attention away from their activities and giving ludicrous explanations to snoopy tourists.
I picked an address based on numerological significance to the campaign (23) [As in, Warehouse 23 of GURPS fame] and a street at random on Manhattan Island. So it’s 23, W 11th St. Currently the address of a Gary’s Restaurant, I believe, according to the limited data I could find online.
Not set in stone, though. I do want to keep the 23 part.
I don’t know New York at all. 
Ah. Well, then, obviously you need a takeout menu from Gary’s thumbtacked to the wall next to the phone.
Every superhero headquarters has got to have a meeting room with a unnecessarilly huge round table with the team’s logo emblazoned on it.
Also, the table must have chairs to accomodate the various sizes of the characters for instance, if one of the team members can grow to 25’ high, there must be a chair for him to sit in that accomodates 25’ high. For characters who can shrink to tiny size, you must have tiny-sized chairs that float a foot above the table. (You can’t expect them to, y’know, assume normal human size and sit in normal-sized chair.) Each seat must have a special logo that identifies the character who may sit in that chair.
No self-respecting superhero would have a headquarters that didn’t contain a gateway to some alternate reality-type dimension (the Negative Zone, the Phantom Zone, the Hellmouth, Dormammu’s Domain, Earth-2, etc.)
A couple of other things:
Material Safety Data Sheets for the various kinds of alien slime, extragalaxial moon rocks, superaccelerated postmutated radioactivoids, and so on. Safety first.
Training videos for new hire superheroes. (“Gallant reads suspects with courtesy and reads them their rights because everyone should be innocent until proven guilty. Goofus assaults them with a blunt instrument and rifles their clothing for valuables.”)
Note from the landlord about the broken toilet in the men’s locker room. Part is on order, will install between 8am and 4pm Wednesday. Leave door unlocked.
Put it in a waterfront warehouse, leading out into the Sound.
Paint a name on the building: “The Hidalgo Trading Company.”
Always nice to have a sense of history, & respect for the Greats, I always say.
Bosda, in my superhero universe there is Abandoned Warehouse Realty. They specialize in abandoned warehouses with anonymous decoration which can be used as a secret base of operations by smugglers, villains, the FBI, secret government task forces, etc.
So there should be a sign out front of their building: “Public Land Use Notice. This old abandoned warehouse will be torn down 10 October 2004 to make way for a brand new abandoned warehouse. Public hearing to be held at…”
And yet, you did not grasp the true meaning of my post.
How sad.
You are not yet a True Geek Master, young Fish.
Nope. Not a thing. Wait, wait… no, nothing. 
One thing not mentioned is the business front.
Where is the money coming from to pay for the 1920’s style Death Rays and all the other gadgets? What is occupying that big building over there- “people go in and out, but no one works there” doesn’t work well as a cover for a secret base. Who cleans up after the heroes (or the villians) trash the place?
Like the old British show UFO, a group could use a film studio as a cover to allow them access in ways that other businesses wouldn’t. Big equipment could be explained away as sets or props, and props could be built for the Mission Impossible style deceptions. The studio could actually make money, and the better thay did the more $ would be available to fight crime (or whoever.) Superheroes could cameo or do stunts- cover IDs would be matter of course. The props department could also forge the papers.
You could add a little humor or tension by having the director or producers working there make life a little difficult for the hero team, with temper tantrums and the rest. Especially if the heroes’ activities are being done in secret.
•Interrogation Chamber.
The layout of which can vary, depending on the morals and ethics of your team. (i.e., whether they’re willing to kill or maim a villain.) You may want to include interrogation equipment and methods that won’t actually cause harm, or you might want to be set up to perform outright torture. The CIA’s Interrogation Manual might be of some help for outfitting such an area in your base.
You might use real-world interrogation equipment—Sodium Pentothal; sensory deprivation chamber(s); a strappado, etc.—or you might be able to use more science fictiony devices, if they’re available in your heroes’ universe—truth rays; agonizers; hypno-rays; a telepath, etc.
•A Crematorium.
Even if you don’t set out to kill your enemies…if they do end up dead, you should probably incinerate the bodies as thouroughly as possible, to keep them from rising from the dead very easily (if at all). If your base uses some sort of nuclear fusion as a power source, you might use reactor plasma instead of natural gas to burn the body. (Unless the villain gets his/her power from high energy plasma, or something.)
In fact…you should probably make provisions the shoot the body in the brain, dismember it, dunk it in a vat of acid, and then burn it. You can never be too safe.
•A Deep-Freeze Storage Bunker.
At the very least, it should be a secure freezer for storing tissue/DNA samples; viruses and bacteria; semen and ova, etc. But, if your universe allows it, you might have a setup to cryonically store whole human bodies for future revival. (Or just as a secure form of detention.)
•A chamber that can suspend things in a magnetic field, in a vaccum chamber.
This’ll come in handy, sooner or later.
•At least one room shielded against telepath, magic, and/or listening devices.
Just keep a log of who enters the room, and when.
•A section of the Medlab that has tall, glass, liquid-filled tubes for keeping bodies in.
They could be life-support tubes for growing clones, treating severe injuries, etc. Or you might fill them with alcohol or formaldehyde, to store corpses in. Or you could make 'em into huge fish tanks, to brighten up the place.
•A specially reinforced chamber for detonating explosives in/defusing bombs in.
You can probably use it as a water tank or something, when you aren’t using it for bombs.
•Private movie theater.
Can double as a briefing hall.
•An escape tunnel linking into the NYC subway systems. Maybe you can even use a fake subway car, for a covert exit.
•A hyperbaric chamber.
Attached to the sickbay/medlab. Can also be used for interrogation, a la Tom Clancy.
Of course, you’ll need some sort of secondary backup base, in case the main base is compromised or destroyed. You can probably put it somewhere out of the way, like the Catskills, or Newark. You can probably have a bit of fun with designing the backup base by having it be basly outdated, and poorly maintained. (Like filling it with really old computer equipment, or something.)
And remember to take some tips from the pros.
Surveillance room! Surveillance room!
Wow, you’re lucky I got here as soon as I did. As it were I saw no less than three supervillains scouting the streets, hoping to find superhero bases without security systems. Swear.
And while you’re at it, how about OnStar for the Batmobile?
Oh, Yeah.
We need a garage for the supervehicles.
I thought this list was getting a bit out of hand until I remembered you said this was underneath a New York City block, not merely a building. So we have quite a bit of space to work with. This base exceeds Lex Luthor’s in the first Superman movie by a factor of ten.
Since its privately funded, its gets a --what? annual appropriation for its operation? You’ll need your budget director, accountants and requisitions approval person.
I would caution against making this place TOO big – it would need waaay too many people for maintenance, and that kind of traffic is just too much to keep plausibly quiet and unnoticed (although I grant you there are ways around this.)
I keep thinking, “Noir, noir… how to make this darker?” Here’s another suggestion, borrowing from Ellis’ Planetary… satellite operatives. Homage consultants. Analogued free agents of this organization who maintain their own smaller bases, mixing and matching whatever elements we’ve brought up that don’t necessarily work at the main base but would fit in perfectly with the personalities of other operatives.
This gives you an excuse to have your heroes move around different areas of New York, interact with different people, see different architecture, meet other agents with a starker worldview, etc.
You wouldn’t use these people outright, just analogues.
Batman. (A retired version of the 1930s original, who killed with a gun)
Shang Chi. (Marvel Comics’ master of Kung Fu, son of the evil Dr. Fu Manchu)
Dr. Stephen Strange.
Baron Winter. (For his house, which is not tethered in time and can be used to access different eras.)
Madame Xanadu.
Kent Nelson’s Dr. Fate.
Doc Savage. (Complete with Hilgado sign.)
Blade.
Father Lankester Merrin (the elder priest from The Exorcist)
Lucien the Librarian, from Gaiman’s Sandman… as keeper of the Ultimate Occult Library.
Perhaps some more background is in order.
The team is funded by a “mysterious benefactor” - though the NPCs that are longtimers when the PCs come on board have met the man, and they each have psychic powers of one sort or another, they know little about him. He has people call him Jack, he appears to be in his sixties, and he apparently has nearly bottomless piles of money.
The team is not secret. It is an investigative firm - sometimes their missions come from Jack, and sometimes from folks on the streets wishing to hire their services.
One of the team members is qualified to run just about any sort of machine available. His power is a link to the human collective unconscious, and he can duplicate the skills of anyone within about a mile. Soemone gets shot, he calls upon the medical training of every doctor within that radius to form an amalgamated skill, and goes to work.
The headquarters is strategically located under a school for child prodigies, to give him access to more vast potential.
His codename’s Meme.
Can’t believe no one has mentioned this yet. (Or if they did, I missed it)
You HAVE to have a terminal with superfast access to the SDMB. You know… in case you have to ask a question.
Band name!
Threads like this are what makes the SDMB great.
You forgot Rorschach (Hurm). And maybe a Van Helsing or two (either Dr. Van Helsing, or Jackman’s Van Helsing). Or Deckard from Blade Runner. And maybe some anime organizations like Hellsing or The STN-J/“Solomon”.
Oh, and if this is to be a noirish secret base, remember to include…
•Plenty of ashtrays; some built into control and computer consoles, as needed.
•Some big, slow-moving industrial fans to stream light through, and/or cast shadows on the floors of dimly lit passageways. (Figure A.) (For reference, see the collected works of Ridley Scott.)
If your team will in fact be working with the occult, a sacred space might be useful, for casting enchanments or detaining any creature that would be effected by holy places (ie: vampires in some mythologies).
Since you’re already underground, you might put in some catacombs. A place where the bodies of past operatives could be both honored and protected from harm/re-animation.
A Holistic Com Room. A place to recieve transmisions from outerspace or the dead or the past. Speaking of the past, hows about a time portal. Something severly limited, of course, but for use in last ditch efforts or only with specific points and rules.
I know people have mentioned fabrication facilities, but I’d go a step further, if your universe allows. I’d have a machine that, given the materials and skematics, can make anything. That way whether you need a ruby sunglasses lens for Cyc, a prosthetic leg for a Chirago demon, or a O-ring for a 57 Chevy, its doable, provided they know how to make it (which given the ability you listed for one of the PC’s should be cake) and can find the materials (which would be the mission/quest part of the process).
The Library, mentioned by several, should also contain vast media holdings from videos to music; computer games to television shows. You never know what you’ll need to look up, and even in the age of the internet, there is no substitute for a well stocked library.
And, since you mentioned Angel as and analog of your team, I insist on a Karaoke Lounge.
