The question assumes Pauline authorship, a point on which scholars are divided.1 It must be noted that I do not take Paul to be the author of Ephesians—I regard this epistle as pseudonymous. This is important for understanding the approach I will present because it begins with the observation that the author of Ephesians modeled this epistle after Paul's epistle to the Colossians (and also 2 Corinthians and Romans to an extent). This dependence on these (largely undisputed) Pauline writings affects how this passage should be interpreted. The author of Ephesians is a Hellenistic Jew (pseudonymously) writing in the Pauline tradition but applying Paul's thought to the situation of the Christian church in his own day (most likely the late first century).
Vv. 14-19 are one long sentence in the Greek text and appear to be part of a prayer. For the sake of brevity, I will avoid discussing the entirety of this passage to focus exclusively on the question asked, but it must be taken as a whole to be fully comprehended and must be understood with the backdrop that the author is adapting other Pauline writings, particularly the epistle to the Colossians.
... ἵνα ἐξισχύσητε καταλαβέσθαι σὺν πᾶσιν τοῖς ἁγίοις τί τὸ πλάτος καὶ μῆκος καὶ ὕψος καὶ βάθος....2
"... that you may have the ability with all the saints to grasp what is the breadth and length and height and depth...."
The way these dimensions are listed is problematic—it is geometrically false:
The final fourfold phrase is difficult; it is wrong to speak, as many commentators do, of four dimensions; there are only three spatial dimensions; here width and length are two measurements at right angles in the same plane, height and depth two measurements at right angles to that plane but in the same vertical dimension and therefore indicate only one dimension. The phrase is then geometrically incorrect, a mistake which no educated Greek would have made. It may have been this difficulty which led to the inversion of the two final nouns in [manuscript discrepancies].... A single article governs the four nouns implying that they are to be taken as a unit and not interpreted separately; [the author of Ephesians] may be using a pre-existing formula. Since actual physical measurements cannot be in mind the phrase must be interpreted metaphorically.3
Knox (along with many other scholars)1 believed that the author of Ephesians heavily borrowed Pauline language in this passage (particularly Romans 8:39 in the specific phrase of interest):
He prays ... to the Father ... that God will so strengthen his readers in accordance with the riches of His glory (the rhetoric of Colossians) in the inner man (2 Cor. 4:16) that Christ may be able to dwell in their hearts by faith (the commonplace of 2 Cor. 6:16), and that they may be so rooted and grounded in love (Col. 2:7) that they may be able to "comprehend" with all their fellow-saints the full dimensions of the love of God. The writer's faithfulness to his model is responsible for the addition to the normal three dimensions of Greek geometry, length, breadth and height (or depth), of a fourth of depth (or height). His reason for the insertion was that he found both "height" and "depth" in Rom. 8:39 and did not understand their meaning; after all the Pauline circle was not really interested in astrology; Paul had used it and then let it drop after the normal habit of midrashic exposition. The writer found it in his original and it suited well with his taste for a lofty style of writing; consequently he has inserted it here.4
This strongly suggests that the fourfold dimensions must be understood metaphorically rather than as literal measurements. But what was the referent of this metaphor?
That they may have the ability to grasp... what?
A genitive referent is expected yet none is provided. Is this formula used elsewhere in contemporary literature? Best, in his commentary on Ephesians, finds no clear support for such a fourfold dimension reading in literature contemporary to the epistle's composure.5
According to Best, some of the suggested ideas for potential referents among scholars include the following, with a summary of Best's refutation of each sub-indented:
the use of spatial imagery to express God's omnipresence
- While similar language does occur in the Hebrew Bible, these specific words are not used and the context is generally non-metaphorical (usually specific references to heaven, earth, the sea, or Sheol).
a Stoic metaphor for coming to understand the greatness of God
- In some philosophical thought, in particular that of Stoicism..., the soul is thought to walk (metaphorically) in heaven and, by seeing and understanding its dimensions, to understand the greatness of God.... Yet we never encounter the fourfold formula and classical writers would probably never have thought of four spatial dimensions.6
metaphorical language for the magnitude of the cosmos
- This has no precedent in the context of the passage and the author likely would have ended the phrase with τοῦ κοσμοῦ if this was intended. However, it may fit into the idea of comprehending God's love from a Hellenistic perspective.
use of a familiar Gnostic magical formula to convey God's power / omnipotence
- This fourfold formula has been identified in a fourth-century Greek magical papyrus, but it is linked with "two other nouns, φῶς,αὐγή, which are not spatial, so that there is actually a sixfold formula and therefore no proper parallel. [Commentators have attempted] to overcome its late date by supposing that texts existed prior to Ephesians which used the phrase, but [they provide] no evidence to support this supposition."6
the measurements of the heavenly city / new Jerusalem, or a continuation of the temple imagery in 2:18-22
- This doesn't fit the immediate context, but it is plausible that it is a revisitation of the temple imagery in 2:18-22 being used as a metaphor for the people of God.
metaphorical language for the immensity of God's wisdom
- This is plausible and overlaps with the idea of 'knowledge of God' (or of God's love).
metaphorical language for the greatness of God's love
- Best and Knox, along with the majority of commentators,7 support this interpretation. The immediate context of v. 19 strongly supports this option, and it also fits with the context of the Pauline language that is likely being borrowed here from Romans 8:39).
Knox agrees while broadening the understanding to incorporate how the author and his audience would likely have understood the text. His observations serve as a helpful conclusion to this answer, tying together several of the above-mentioned interpretive options.
Paul had written (Phil. 3:12) of "grasping" God as God had already grasped him, in language in which the thought of laying hold on God was coloured by the thought of "grasping" the goal in a race (1 Cor. 9:24). The writer of Ephesians seems to have missed that colouring and treated the language of Philippians as if it meant "comprehending" God with the understanding in the sense of "knowing" Him as in 1 Cor. 13:12. The change in meaning was natural, for the difficulty of finding God, of which Plato had written in the Timaeus (28c), was one of the most hackneyed quotations of Hellenistic literature, especially in Jewish literature, which used the quote to prove the need of a special divine revelation, such as it alone possessed....
In the present passage the rhetorical dimensions suggested the vastness of the love of God; it can indeed be "comprehended", yet it passes knowledge (Rom. 11:33, 1 Cor. 2:9). At the same time the thought may have been coloured by the common belief that the "mind" can ascend to heaven and "comprehend" its dimensions or the variation of the theme in the form that the full "comprehension" of the cosmos carries with it the "comprehension" of God.
In any case the love of Christ cannot be fully understood; it surpasses knowledge and is better than Gnosis. It is the only power which will bring the readers to a completeness in which they can contain all the fullness of God (Col. 2:9). The transition from Christ dwelling in the heart to the "comprehension" [with a suggestion of occupying], the full dimensions of the divine nature, and back to the thought of being filled with a divine pleroma, would be abrupt if the passage were not largely a cento of Pauline phrases and if the transition from the language of the soul's ascent to God to that of God's descent to the soul were not a common feature of popular theology.
The writer has thus established the plain Pauline system of the love of God as manifested in Jesus as the supreme mystery and the highest Gnosis.8
1 Percy, passim; J. N. Sanders, ‘The Case for the Pauline Authorship’, Studies, 9–20; D. E. Nineham, ‘The Case against the Pauline Authorship’, Studies, 21–35; H. J. Cadbury, ‘The Dilemma of Ephesians’, NTS 5 (1958/9) 91–102; L. Cerfaux. ‘En faveur de l’authenticité des épîtres de la captivité’, Littérature et Théologie Pauliniennes, Bruges, 1960, 59–71; R. Kasser, ‘L’autore dell’Epistola agli Efesini’, Protestantesimo 17 (1962) 74–84; J. Murphy-O’Connor, ‘Who Wrote Ephesians?’, TBT 18 (1965) 1201–9; J. I. Cook, ‘The Origin and Purpose of Ephesians’, RefR 18 (1965) 3–18; Van Roon, passim; Caragounis, 35–56; J. B. Polhill, ‘An Introduction to Ephesians’, RevExp 76 (1979) 465–80; G. A. M. Vleugels and J. C. Coetzee, ‘Onderzoek naar de synoptische relatie van de brieven aan de Efeziërs en aan de Kolossenzers’, In die Skriflig 22 (1988) 37–46; M. D. Goulder, ‘The Visionaries of Laodicea’, JSNT 43 (1991) 15–39; J. H. Roberts, ‘The Enigma of Ephesians - Rethinking some positions on the basis of Schnackenburg and Arnold’, Neot 27 (1993) 93–106.
2 Eberhard Nestle et al., Universität Münster. Institut für Neutestamentliche Textforschung, Novum Testamentum Graece, 27. Aufl., rev. (Stuttgart: Deutsche Bibelstiftung, 1993), Eph 3:18.
3 Ernest Best, A Critical and Exegetical Commentary on Ephesians, International Critical Commentary (Edinburgh: T&T Clark International, 1998), 344.
4 W.L. Knox, St. Paul and the Church of the Gentiles (Cambridge: Cambridge University Press, 1939), 191-2. Available on Archive.org. Footnote #4 on p. 191 corroborates the notion that Greek geometry clearly understood three dimensions.
5 Best, 344-6.
6 Ibid., 345.
7 Including A. T. Lincoln, Ephesians (WBC), 207-13 and the NET translators, as noted in the question.
8 Knox, 192-3.