Assuming one had a couple of students and a place to train. What would be considered the minimum essential training equipment to have? Let’s also assume that the place is rented and nothing can be left there. Everything just be portable or easily removable after each session is done.

What items would you get and why?

Depends on what you plan to do, but for me, it would be sparring gloves, mouthguard, groin protector, and a pair of Thai pads, or some other kind of striking mitts. If you could do mats, that would be ideal, but for bare minimum, this would cover the vast majority of partnered training. You have the safety equipment to spar with, and pads you can use to work your techniques and combinations with full power, to make up for the fact that you won't be going full-force with everything in sparring. Throws and takedowns will have to be done with control, though, or skipped entirely, without mats to fall on.

Depends on what you want to teach but Noah's list covers it. Heck if you only want to teach kata and kihon you wouldn't even need anything, just a clear space.

I was thinking about striking pads, mitts or shields. Things that could easily be carried every time they’re needed as nothing can be left/stored at the training area due to use by other groups for different purposes.

Realistically, all you need is the space. If you have some money, striking pads, shields, and loaner mitts would be ideal. When I started karate in '02 my teacher was new to teaching and didn't have anything built up for it. He used the dues he got from us to buy pads and mitts. Eventually, he got enough students to rent out a space at an MMA gym that allowed us to use their bags, bobs, and the like.

I agree on the thoughts about some striking pads of some sort. As a TKDer, I always enjoyed kicking on those clapper pads, too. Big, thick kicking shields are great if you can't have a heavy bag.

A Kicking Shield or BOB and some focus mitts, at the barest minimum. For the most, practically any MA training equipment is portable, if one thinks about it.

if money isn't an issue and it is solely storage then you can also get rollaway mats that could be brought too and from training. That would allow for some ground work/throws/takedowns to be practiced.

Would agree with the other comments on focus pads, gloves etc.

I've also trained in similar places before and we were able to get access to a locked cupboard or add a lock box and leave mats etc in there. Decent padlocks and a sturdy storage box did the trick.

I think Thai pads (or pro force hand pads if you need cheaper), and kick shields would be the bare minimum for the instructor. If there’s sparring, I’d make the students get their own gear.