The Night Nurse, the Day Doctor, the Twilight Therapist....

Inspired by a mention here (look for Atrox Morbus), here's one that's fine-tuned to a modern superhero campaign....

The Walker Clinic

It's an open secret among the crime-fighting community...if you're injured, you can get treatment at the Walker Clinic. The only supers who can't are those who can't be teleported.

You have to go through two rooms to get to the Walker Clinic. Each room is a teleportal, capable of teleporting two tons (there was a little accident early in the Clinic's history, which is why the weight limit is so high). The first portal takes you out of the country to your favorite country that has no rule about reporting gunshots or injuries from superpowers; the second portal takes you to the clinic itself, which is...well, somewhere else. The first portal is so that lawyers cannot claim that the clinic is bound by the rules of the campaign setting; the second takes them to the actual clinic. This is why the police (if they know about it) don't go there.

The Walker Clinic doesn't have huge amounts of technology. They seem to have high-tech diagnostic equipment. If they can't fix what is wrong with you, they can probably describe what is wrong with you. Their surgeons are excellent, and their healing times and recovery rates are excellent, but some people die.

They don't promise results. (No hospital does.)  They're not so good on magic....but they do have the names of some sorcerers you can consult in case of a hex or curse, and they can sometimes provide symptomatic relief.

Insofar as the rules go: No fighting. Obey your medical practitioner.

The Walker Clinic charges a flat fee: One hundred dollars or the equivalent in your local currency. It doesn't matter if the treatment is major, minor, or inconclusive; the payment is one hundred dollars. If you truly have no money, payment is a favour, probably {but not guaranteeably) acceptable to your morals.

No favours have ever been called in....or if they have, the people who provided the favours aren't talking.

So what is the Walker Clinic, really?

Option 1

They are the advance scout for an alien invasion. The invasion might not happen, but the cost of setting up a clinic like this is trival compared to the cost of getting to Earth. (The alien invasion might be from another dimension.)

By setting up a clinic, they can scan, analyze, and index superhumans. If they have to perform surgery, that's a bonus, because every person operated on gets a hypoallergenic hyperwave-activated package of nanites that will either kill them (if they've never visited before), cripple them (for those who are "normal" humans, like Batman or Daredevil), or remove their powers (if they've already been analyzed and indexed).

From a story point of view, sooner or later that invasion is going to happen.

Option 2

They are the legacy of a doctor-superhero who wanted to help people, and if it were associated with him, he knew that supervillains wouldn't show up. The teleport gates and high tech diagnostic are all repurposed super-technology that he has acquired over the years. Favours with a don't-tell rider get used to fix the place. It is paid for by Atlantean gold or some such--those old-time heroes had a lot of wacky adventures and money never seemed to be a problem.

The founder has passed on, finally, and the new manager is implementing some changes, which will affect our heroes. The government wants very much to have access to that facility or their index. (The technology they don't care so much about.)