So, what exactly is it?

I knew the train was made by Varney, now... but I always thought Varney was a company known for making high quality HO locomotive kits, especially steam locomotives, during the postwar era. What was this cheap, simplistic model doing, carrying the same name?

Searching "Varney F3" didn't yield any results, though the plastic shells looked to use more or less the same mold.
Then I tried "Varney train set".
And there, in the results on Google images, I managed to find it-
with a box, that called it the "L'il Raskal".
I then searched for that- and found an advertisement being sold online, which both dates the train, and also explains why on earth it was designed the way it is.
  
So, there we have it... Varney, marketing realistic scale models, decided to produce a train designed with the explicit intent that it was not a model, but a toy, and it would survive the worst misuse and abuse a child could throw at it.
Looking at the equipment I have, I question if these trains were really as tough as they claim, though I have no intent to attempt to break mine (any further than the damage they already have sustained in their previous lifespan). It also seems likely the "High Impact plastic", however tough it may have been back in 1958, is a fair bit more fragile now. The couplers do seem to be made of a flexible plastic, which I'm sure lead to very few broken couplers unless the user had the specific intent to attempt to damage them.

I'm looking forwards to getting this odd piece of equipment working again- there's something very interesting to me, in trains like these, which depart from how a model typically "should" look, for the sake of hitting a price point, or in this case, level of durability.

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Even if someone took these shells and mounted them on the 2 axle chassis, those chassis are very peculiar indeed!

I stand corrected - Ebay has two print ads right now for the lil rascal set.  There is indeed a caboose out there, along with a power pack of sorts that may be at the antique shop but you missed it because you didn't know it looks like a flashlight tube,

I found all there was

I will go back again and check, but these were on a shelf, with some toy cars- I say "pile", but it was more like a large handful of HO scale train cars gathered together haphazardly sitting on the shelf. I looked all about the assortment, and these were all the "L'il Raskal" items there. I don't recall seeing the battery tube, but you're right, I don't think I would have seen it for what it was, until I had seen an advert for the set. 

It's quite strange indeed, and there's no doubt about it being original- even before seeing any information on what I had, I could tell they'd been built this way- the molds for the freight cars all were made specifically to accommodate the metal frames, with flat, undetailed bottoms that feature locating nubs for holes in the metal frames. I also doubt that your average hobbyist would fasten the metal frames to a plastic body with rivets- what really had me confused, is I'd expect this to be some "no name" import from Japan or another east asian country. And the locomotive has "Varney" molded into the plastic part of the chassis- the metal frame is split down the middle, and riveted to the plastic part, so the frame sides serve as the electrical pickup from the wheels.

All in all, it does have the look of someone having slapped a few shells on top of some cheap, simplistic 4 wheel chassises. Out of curiosity, I decided to punch the retail price into an inflation calculator. According to that calculator, the retail price of $9.95 USD is equivalent to around $90 USD in 2021. Personally, for that money, I think there was better trains to be had in 1958. I'd imagine Varney also found it to be true, since it doesn't look like they ever expanded the product line beyond the one set, and the various add ons in the advert (I'm assuming they made them, but I haven't seen any pictures of the add on cars yet).
Still, I think they were onto something with those flexible couplers- considering how easy it was/is to break the normal kind.

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I regard the XF2 as one of the single most detrimental things to hold back this hobby...even if they were everywhere...it would be appropriate in this set!

1958 prices

I bought an Athearn GP 7 in 1957 , it cost $7.95 IIRC . I still have it , converted to a newer drive .If I ever build another HO scale layout I'll add sound and DCC to it :> ) ...DaveB

The ad says "super detailed

The ad says "super detailed scale bodies".  It just goes to show you how much the definition of super detailed has changed over the years.  That thing belongs in a museum in the section on evolution of model trains.