Hcfyk

The word "shop" seems to behave like a non-count noun in phrases like "set up shop", "shut up shop" and "close up shop". There's no article ("a"), no plural ending ("-s"). Dictionaries, such as Oxford and Merriam-Webster only list count usages and singular usages (as in "the shop"), so this idiomatic usage of the word "shop" appears to be very unique to these phrases, because the word doesn't seem to have common non-idiomatic non-count usages at all. In comparison, take the word "business", which has both common non-idiomatic non-count usages ("get down to business", "bad for business") and count usages ("small businesses").

So why does the word "shop" have this non-count usage in the phrases above? Etymologically or historically, was there any point at which "shop" meant "business", but such a non-count usage only survived in the idioms, while only the count usages survive elsewhere?

Merriam-Webster provides a dictionary definition for this phrase under its definition of set up (search for "set up shop"):

set up shop

: to start a business or activity in a particular place

As a set phrase (or idiom), it's an expression that isn't defined by any of its individual components, so analyzing the syntactical role of shop doesn't work well in this specific instance.

It's possible to break with the idiom and say that you're going to set up a shop or set up the shop, but that would be in relation to a specific, singular, situation, and it wouldn't have the same meaning as the phrase itself.

This is not an authoritative answer--which would have to be blimey, I don't have the foggiest--but a barely educated guess.

English shop has a cognate in Low-German Schuppen, "shed", think tool-shed, work-shop; cp. Bude "abode, house, building; (coll.) construction work company". This is homophone with Schuppen "fish schales", originally a rare kind of uncountable pluraletantum, if I read that correctly (thus die Schuppe would be a back formation by analogy). I haven't seen these related by the root, but if I were pressed to find a connection, I'd say that roof, cover is the primary symbolism for building ("under my roof, under my table") and that fish scales are significantly reminiscent of roof-tiles. So, if shop was ever understood as a non-count noun, that must have been a while back, perhaps in an isolated dialect. It's odd at least insofar the modern genus is maskulin, der Schuppen, while -en usually implies die (plural noun) or das (nominalized verb).

Incidentally, Frisian was isolated for a long time, though I wouldn't take that alone as an argument. Rather compare shovel, Ger. Schaufel, also Schippe. Again there's something about the form. Comparing Scheibe, Schicht, Schindel or Schiefer would maybe go a bit too far though.

There's also what looks like a verb schoppen, chiefly in the noun Frühschoppen "early bar opening hours, a drink in the morning", and now it becomes uncertain--if it wasn't before. A Schoppen is a measure of liquid, half a pint, related to Fr. chopin, and eventually related to schöpfen, scoop. The problem is that Schöpfer "creator, god" draws the attention of this word. This in turn may relate to schaffen "to work, succeed, create, scape", -schaft "-scape, -ship, -hood" (Landschaft, Herrschaft, Gemeinshaft, Nachbarschaft ...). There' are Schöffel, Chef, Schaffner, Schäfer, that relate more or less. Geschäft means "business", geschäftig "busy".

There's also shaft, that gives me the peculiar image of a shop-keep opening the window lid (the shop?), proping it up with a shaft; Alternatively pulling the blinders aside (Gardinen is another pluraletantum). Cp. Schaufenster ("shop front", literally "viewing window" or "show window"), Schaubude "fare establishment", Ger. Schotten, Luken, Klüsen (closures in windows or ship-pipe-works), and very significantly Laden "shop", Fenster-Laden "window lid" (those on the outside; another unusual maskulin -en noun); also see En. ladle, Ger. Schub-lade "drawer" (as if a lid that is shoved), laden "to load", and perhaps Umschlagplatz "market?", umschlagen "to flip, to swap", Klappe "lid, that which flips and clips", verklappen "to distribute [waste]", market, probably via Etruscan merx, in my humble opinion related to PIE *mey- "to change". Which Laden was first, and how it relates to shop I can't say, but it looks like a loaded semantic correspondance.

PS: The image of a shop-front propped up like that of any fast food truck came when I searched for scappa--pretty much into the blue--which is Italian and relates to escape, which has an oddly uncertain etymology (to cloak, really?). This reminded me of fire-escape, ancient Anatolian buildings with the entry through the roof, and thus outlets over fire places in all kinds of buildings, thus window. shaft is an after thought.