I’d look into HVAC soundproofing systems. We use materials to baffle the air movement as it passes through the ducts. The baffles keep the noise in the heating unit from entering the room.

And what is your budget for this? $50? $50,000?

@RonBeyer Any solution that fits the criteria (with some flexibility) is acceptable. I imagine it can be expensive but there can be clever and cheaper solutions.

Build an anechnoic chamber with a door.

@MonkeyZeus Yes I know them but it is not the solution I want. It is difficult to use them for long periods also.

@RonBeyer I know anechoic chambers but I want a permeable solution. It is the main part of the question. Is it possible to create a permeable anechoic system?

Yes, they have to get into/out of them somehow. They can be vented...

@LeeSam I don't know much about HVAC soundproofing systems. Is it permeable? If you know the details, you can provide an answer. However, it seems like it is contained in a soundproof room: "The goal with these HVAC soundproofing treatments is to contain the noise to within the room and protect against the bleed of the noise out." However, I couldn't find info about baffling the air movement.

Look into how the sound room in a recording studio is isolated. That kind of room is designed to keep outside noises out so they don't get into the recording. Yet it is permeable, too, as there are passage doors for people and instruments. You will be looking toward the higher price range for this, but the results should be good. Note that if by 'neighbors' you mean 'apartment neighbors' (ie you rent) you'll be limited in what you can do. Consider making a smaller "sound room" inside one of the rooms you have available.

Perhaps we need to clarify the condition "I can walk through the sound barrier." If this means opening a door, then fine, we can deal with soundproofing doors [& walls & windows]. If it means literally nothing to impede, just a walk-through 'hole', then see my answer below. Soundproofing requires mass. Nothing else is a substitute at anything below 250Hz or so. Top end is easy to filter out, low end is not.

@Tetsujin I thought it is clear and you understood it. It is a hole. It is a doorway without a door. It is a sound barrier (soundproofing) for a hole/opening basically. I thought there could be solutions with active noise cancelling or through certain sound waves/electromagnetic radiation etc. Also soundproofing this hole might be part of a bigger system.

Active noise-cancelling requires sub-millimetre precision. Achievable in a headset or mobile phone, not in a doorway. You cannot prevent noise coming in an open door, the same way as you cannot prevent water pouring down a plug-hole.

@Tetsujin I actually asked in Worldbuilding Stackexchange first but they said it is more for Sound, DIY or Engineering Stackexchange. But if we are sure that there is no solution, that can be the answer. However, a solution with as much as possible noise cancellation is ok too (to cancel above 250Hz possibly as you said). Maybe someone is working on this technology, who knows :)

This is beyond our current knowledge & application of physics. Chaos theory will win until someone invents the 'silence field'. So far, that's Sci-Fi ;))

So this is not a DIY question for a specific room in or around your home, but a "world building" question. Otherwise, please specify the exact room, its dimensions, doors/windows, how much space you have/need, activity in the room, the types of sound to be kept out/in, your budget and level of skill with drywall/framing etc.. and other constraints. Pictures and drawings would be great too.

@P2000 They said this question is not good for Worldbuilding.stackexchange also and I deleted it myself there. I didn't know if it is achievable or not. Many questions on here don't have the details that you listed but my question is very detailed as it stands. It is an average door like all houses have. I don't know if I should ask on Sound or Engineering stackexchange.

Try electrical engineering (EE). Your question is in my professional field of expertise, but you really have to be more concrete about what you are trying to accomplish. At EE we don't need your drywalling skills, but we do need a lot more context. "I have a ... and I am trying to ... How should I ... so that ... " is a good start.

@P2000 - I'd love to know what you have in mind for this. After 45 years or religiously closing the studio door, to discover there's an electronic 'cure' for musicians apparently 'born in a field', other than just a mechanical door-closer... ;)

@Tetsujin I have 37 years experience with force fields and flux capacitors, and once I know what the OP actually needs, I'll know for sure its neither of the two...