Why do people say "quote unquote something" and not "quote something unquote" ?

It's "quote unquote something" because most people who "quote something often forget to unquote afterwards.
- I see what you did there.... 🤣
- And that breaks the processor and you have to reboot your listener and it's such a paaaaaiin.
  
  Broke my brain, had to read 4 times to understand

If you are quoting a word or short phrase you use this form to make it quicker and easier for the listener to understand.
If you quote a long section, saying "quote,
<long quote>
, unquote." is common and accepted.

It's the verbal equivalent of quotation marks done as a hand gesture.
- Yeah—I think the canonical usage is to hold up your fingers as you say “quote unquote”, then lower your hands when the quote is complete.
  
  One hand for quote, the other for unquote

They are just doing the autocomplete verbally, like when you type an opening quote and the end quote goes in automatically but the next thing you type goes inside the quotes
- Plausible for programmers, at least
- This is a solid take
- I hate this functionality.

Because "quote unquote" is done for a laugh typically and "quote unquote" sounds funnier and more pleasing to the ear.

this is one of those things that I have wondered about for so long that I forgot to wonder about it
- How would I even know where the quote ended

Professors and engineers, in my experience, tend to say "quote... the thing... end quote". Regular people on the other hand, are lazy, inconsistent and generally oblivious to whether or not they're being ambiguous.
- Regular people on the other hand, are lazy, inconsistent and generally oblivious to whether or not they're being ambiguous.
  me_irl
  
  o7

Unrelated but until a month ago I've been saying "quote ON quote" until I saw it actually written 😂🤣
- When I was younger I said quote END quote.
- As a homeschooled kid, I usually had the opposite problem. Mispronounced so much shit.
  
  Archipelago.

Tangential, but I don't understand why in American English you feel the need to say the word quote at all. In UK English we just use intenation.
- Can I quote you on that?
- It's useful for when you're quoting someone who happens to use the exact same intonation as you!
- I'm from the UK and I feel like I've heard enough UK English speakers saying "quote" that I had never thought of it as an American thing. That isn't to say that the distinction you make doesn't exist though, just that it may be variable across demographics or contexts.

Because it would be pretty silly to verbally say "quote" "the thing" and them finish of with "unquote" at the end, like some kind of robot.
The whole point of saying it is to clarify that you're quoting something.