My sense is that the answer is "Yes" -- it's a metaphor for a "real place", whatever that "place" might be.
Psalm 23 has a series of shifting metaphors, but in the opening verses the governing concept is that of the LORD as "shepherd", so that the "green pastures", "still waters", and "right paths" of vv. 2-3 are all metaphorical places, of nourishing, of refreshing, of ethical goodness, within that broad conceptual framework in which the psalmist experiences the Lord's care as a sheep experiences the care of the shepherd.
The "valley of the shadow of death" in v. 4 participates in the development of this metaphorical movement. The relevant bit of the verse reads this way:
גַּם כִּי־אֵלֵךְ בְּגֵיא צַלְמָוֶת
gam kî-ʾēlēk bĕgêʾ ṣalmāwet
Here, gêʾ is the word for "valley", and ṣalmāwet the word for "shadow of death". This translation is itself somewhat controversial, or at least difficult. It appears to be a compound noun, but:
- should that "compound" be understood in its component parts? = ṣēl "shadow" + māwet "death", or
- should it be taken as a single concept, "deep darkness", "utter blackness" or the like?
Modern lexicographers and commentators tend to take it as the latter, see e.g. the entry in Brown-Driver-Briggs, p. 853:
In Ps 23:4, the Greek translator, however, took it as the former: ἐν μέσῳ σκιᾶς θανάτου, literally "shadow of death". Even the much more "liberal" translator(s) of Job and Isaiah used this equivalence.
The usage of the word is somewhat broader that Ps 23:4, of course. It appears 17× in the Hebrew Bible (see the entry of BDB for references: the little dagger symbol, †, indicates that all occurrences have been cited in the entry). Note especially the frequent use in Job (8×).
In biblical terms, then, this is consistently evocative language for the place of danger, which is especially the place where God's absence is felt (cf. e.g., Jeremiah 2:6).
Later interpreters took it in different directions. Rashi, for example, explained it in terms of of the "wilderness of Ziph", in 1 Sam 23:13, 28 [as given by Rashi, presumably because they set out David's predicament and escape, but see vv. 14-15 and 24 for mention of the wilderness of Ziph]. This is to take Psalm 23 (a "psalm of David") in biographical terms, and relate this language to a particular episode in David's life when he was under threat of death.
In quite different terms, it is part of the allegorical landscape of John Bunyan's Pilgrim's Progress where Pilgrim faces Apollyon in the valley.
Note: this answer might be regarded as v0.1; I might come back for an edit.