So, there's a local ESI-R that's creeping down into the "how can you afford not to" price range on the list of Craigs. I know next to nothing about these other than they have great body lines (IMHO) and you don't see them every day (maybe for good reason). Other than that, my other observation is the engine bay looks like a nightmare.
Besides the inevitable issues with dragging home a car I don't need or have space for, what would one look for as common issues/failure points? What sort of potential do these have? Am I an idiot for even looking at this? Etc. Etc.
There is a really good FAQ here that should answer a lot of your questions, these guys are one of the only tuners left in the StarQuest game: http://www.racetep.com/tips.html
I still want one and will own one soon, I've just never found one in the right condition when I had the money.
Heads are known to crack, that racetep site has replacements or you can import a "JDM" one from Japan, Australia, or the UK. This happened to a friend of mine.
Primitive fuel injection system can lead to problems if you want serious power but for most people's goals you are good to go. You could import a MPI setup + the "JDM" head from an Australian Magna.
These cars are known for electrical issues. Them being old won't help.
Look for rust, these are fairly resistant but it's still an 80s Japanese car if you are in the rush belt be thorough.
These cars are fairly awesome stock. I love the wheels on the later ones like the ESI-R.
Typical old car stuff will go wrong obviously. It could be anything at any time.
This won't be an easymode car. They were known as being unreliable when they were new. Parts are not easy to source. Near non existent aftermarket. Like Nissan supports their cars from the 80s a bit, Mitsubishi does not at all. Junk yards and the internet will be your friend. They are the top tier of cool though if you really want one it will be worth it.
If you want a cheap 80s turbo car that is easier a Z31 300ZX or the MK3 Supra would be better choices. Or with Mitsubishi a Galant VR4 if you like AWD. Starion is a much cooler vehicle than all of those tho imo.
Vigo
PowerDork
3/7/15 1:46 p.m.
I dont think the engine bay is a nightmare at all. From an ownership perspective, probably the biggest issue is electrical connections flaking out due to corrosion or breaking wires.
From an 'im going to modify it' standpoint, probably the hardest part is making sure you understand how the TBI system works before you modify it. As far as i can tell, most of the starions that were broken in the process of going fast were because of fueling issues caused by a combination of poor electrical connections and not understanding how the system worked before modifying it. 
The actual engine itself is strong and has plenty of potential. There have been a decent number of people who've gotten 300-400 whp on the stock tbi setup (without going mpi, in other words).
I miss my 88 Starion. Had it for a couple years before the rot got to the frame. I never had any problem working on it. http://www.starquestclub.com is a good place to start. These cars can frighten people who know nothing of them. I had always loved them & did the research prior to buying one. The biggest problems I had was probably chasing down an odd misfire which ended up being a faulty distributor.
And its gone. It was an 88 ESI-R for $2,200 with the head swap already done.
They're for sale everywhere for various prices in various conditions.
What to hate:
-Jet valves
-Injector clips go bad (studder and bogs).
-Vacuum leaks (lots of vac lines).
-Etacs faulty.
-Steering wheel column has a bushing that causes it to fall to the lowest position.
-TBI fuel injection sucks.
-Head cracks if you run high boost.
-No real aftermarket.
-Body parts are hard to find (lower airdam are almost always cracked).
-Not a lot of suspension options.
-Mitsubishi clear coat.
-4g54b
What to love:
-Beautiful lines (mmmm box flares).
-Rare.
-Can swap anything in them.
-Fun to drive.
-Easy to work on.
-Super 80s.
-Best looking thing in my garage.
I love my starquest. I can't find myself to part with it no matter what. I've owned both a Galant VR4 and Starquest. Wish I could have both at the same time stock form. I will say the starquest is definitely cooler. 
In reply to yupididit:
I've got a Starquest Lemon's car. It's one of the most fun cars I've driven. Granted, we stripped almost 1000lbs. out of it, but it handles better than any zero budget, homemade parts wearing car has a right to. Stock the cars are overweight, but you could cut quite a bit pretty easily and still have a nice street car. I've thought long and hard about building one into a nice street car. The stock engine isn't bad if you know what it is and play to it's strengths- it's a big four cylinder truck motor with a turbo hung on it. You will pull your hair out if you try to turn it into a high RPM screamer. Instead, shoot for V8 style torque, and it can put a big smile on your face. That said, I'd do an engine swap if I built one. The car begs for a V8. I've also got a Galant VR4, and don't think you can compare the two. The Starquest looks cooler, no doubt, but few live up to their looks. The Galant is cool because it looks so boring, but most would destroy many cars that look much faster.
