For the people here who do not yet work entirely electronically, and still use printed papers & old school folders to organise their papers, do you guys have a system for organising these papers?
I have bought a few folders today and have made a few rough distinctions, but I am wondering how more advanced academics organise their papers once they've read hundreds to thousands of them (and also roughly on the same very specialised topic).
Separate the semantic ordering from the physical ordering. It keeps your organization flexible, since you can assign each paper to several topics and projects, and change the assignments as you see fit.
This can be achieved by using a filing cabinet or folders on the one hand, and a bibliographic database (be it electronic or in paper) on the other.
Label each paper with some identifier. I recommend the date or month, but also a running number will do. The label doesn't have to carry any meaning. Collect the papers with a certain range of labels in one folder. For example all papers with labels January to March go in one folder called "Jan-March 2018".
In your database, each entry should also carry the corresponding label under which the paper can be retrieved, in addition to the bibliographic data, and any semantic information such as projects and topic keywords. Use the database, not the physical location, for organizing the papers by substance matter.
How the database itself should be organized is another question. I use folders for projects and keywords for topics. Creating a systematic ontology or adopting an existing ontology (e.g. the Dewey system) seems overkill. Instead, I recommend creating keywords on the fly ("tagging") and combing through the keyword list once in a while. This will be more difficult with a paper-based organization than with an electronic database, of course.
Personally, I am not keen on organizing printed papers. Even though I prefer to read paper documents, I have never been able to devote enough space for proper organization and have resorted to ad hoc printing of those I really want to analyze in detail (knowing that after a few days after reading I will misplace them somewhere) and on-screen reading of those I really only want to skim through.
However, before the introduction of digital technologies, the storage of paper-based documentation was an art. It was perfected by well-known sociologist, Niklas Luhmann (indeed, his system of filling cabinet was one of the things that he is famous for). He used system of filling cabinets to store tens of thousands of paper documents: 
As far as I know, he used filling cabbinet-based system for storage of a number of paper fiches, used to organize different ideas: abstracts of books, ideas for papers, etc. I am no aware of him using this method to store full papers, nevertheless you may look into it as an inspiration for your problem solution.
Luhmann's system was subject to a number of studies. Himself, he described the system in the following paper: Kommunikation mit Zettelkästen. This seems to be one description of his system in English.
I sometimes use printed papers in folders too, because it's nice to be able to peruse them without aid of a computer, especially while in a coffee-shop, etc. Others here suggest using a flexible ordering system, but personnally I find that I use the folder to refresh knowledge of a particular research topic for purposes of writing papers in a specific area. Here is what I put in the folder (in order):
One folder of papers per for each broad "area" of research (subject to length considerations).
Cover and spine: Cover page and spine page with clear identification of topic, and my contact details on the folder in case I lose it.
My own papers: My own papers on the topic, in whatever order is most logical, followed by a coloured separator tab.
Bibliography and literature for each topic: For each topic covered by the folder, a cover-page giving full bibliographic information on all the papers for that topic (listed in chronological order of publication), followed by the papers themselves, separated by a coloured page. Sections on different topics separated by a coloured separator tab.
Since I use my folders of printed papers to refresh particular research topics, I don't worry about doubling-up papers across folders (though I might if I started tripling-up, etc.).
The system I use is designed to allow me to review/refresh my knowledge of a particular topic (where I have a research interest) in the order in which the research occurred historically. I can also jump to particular papers of interest easily. The folders are also a useful aid to lend to graduate students if I want to give them some background reading in a research topic.
