That said, I personally think that the Z architecture is completely bonkers and I would love the opportunity, like you, to learn it more in depth. My info above comes second-hand from what I've been able to piece together in my own searches.
[0] http://www.hercules-390.eu/ [1] https://www.luxoft.com/banking/core-banking/ [2] https://en.wikipedia.org/wiki/CICS , https://www.redbooks.ibm.com/abstracts/sg247952.html?Open
"Z architecture" isn't all that different from the Linux for MVS integration, so it's not all that bonkers. Trying to understand mainframes in terms of Unix/Linux is totally bonkers, though.
https://ibmzxplore.influitive.com/users/sign_in
Also a bunch of free courses here:
(free registration required)
(EDIT) If you're in the US and meet the requirements (take free courses) you might also be interested in this for (I believe) $120/year:
https://ibm.github.io/zdt-learners-edition-about/
Email in profile if you want to know more.
I then worked a bit with IBM (in Poughkeepsie, NY) getting Linux on MVS working securely before it was released as a product, and I haven't touched a mainframe since. </background>
The Unix world and mainframe world are so different that you couldn't really just throw an emulator on a Linux box and be able to get anywhere professionally with it. Even the lingo is not compatible. There is next to nothing from the Unix world that you could apply to mainframes. You will have insurmountable gaps in your knowledge without full immersion. Talk to a mainframer about buffer overflows, and you'll realize right away that you are on completely different planets.
AFAIK there are still no full MVS emulators for Linux because of the sheer mountain of "stuff" that would need to go into it. You can certainly learn the scripting languages with emulators, though. So there is that. But your lingo and even understanding of how things connect would still suffer greatly if you think of the concepts from a Unix/Linux point of view. They are not compatible.
Having said all that, because there is Linux on MVS, in theory you should be able to use it as a familiar terminal. But then your only access is through getmvs which still keeps you much too distant from the iron to call it mainframe experience. And it is unlikely anyone's going to open up a terminal to the Internet, as it was for me back then, too. There is way too high a financial risk involved.
Things may have changed since then, but mainframes really don't change very much even across decades. I think getting Linux running in a VM was probably shocking to the entire mainframe world when it was first released. Banks gobbled it up though, so it was quite a smart move.
I believe cloud computing has totally taken over and removed any benefit there was in that particular area now, though.
My expectation is that you'd still need to rent mainframe time slices, and go full immersion if you want to get "Mainframe" on your resume. Otherwise you might have to lower your bar a little, and stick to some of the batch processing languages instead.
http://dtsc.dfw.ibm.com/adcd/adcd.shtml
Let's say I were to get an emulator running, or somehow install a mainframe in my basement. Then what? What could an average person like myself do on a mainframe that would be useful and benefit me in terms of the job market?
I can download a Linux ISO and run anything I want to on a distributed system (x86 or ARM PC.) That type of tinkering is how I got my start/interest in technology to begin with. But it's not as though I'm about to process financial transactions in my basement for fun. How do you get started "tinkering" with mainframes?