I prefer sewn binding with nice spines whenever I can.

But for zines, I do this thing I used to do for collecting comics.

• Half letter (5.5x8.5”) zines go in these sheet protectors and into a binder where I can easily flip through them
  • https://www.amazon.com/dp/B07F4677WJ
  • https://www.amazon.com/dp/B083V775DK
• A5 zines get sleeved with these, depending on thickness:
  • https://www.amazon.com/dp/B00BPENIH2
  • https://www.amazon.com/gp/product/B07JRBWSTP/
• The sleeved zines and small books go into storage boxes. The boxes allow me to flip through them like browsing records. I add dividers and organize them by system or type:
  • https://www.amazon.com/dp/B09GNQV658
  • https://www.amazon.com/Short-Comic-BIN-Plastic-Black/dp/B077PPFN35
  • https://www.amazon.com/PENCO-EB013-Storage-Container-Clear/dp/B07XYJHRGK
• Thick/heavy hardcover books stay on the bookshelf

The room I store them in follows archival guidelines for comics and paper documents. 50-65F stable temperatures, and 40-60% relative humidity. Coincidentally, these are my exact preferences for my bedroom.

The 3x6x9 version of these fit zines perfectly and are archival quality. Don't use the magazine holders suggested by the other user.

https://www.gaylord.com/Document/Gaylord-Archival%26%23174%3B-Blue-Grey-Barrier-Board-Pamphlet-File/p/HYB02276

And if you want to take it a step further, these photo sleeves will fit 95% of zines perfectly.