Firstly a quick journey back to roots...
'Just', in modern English usage meaning 'exactly or only', and 'closely or proximately in space, quantity or time', as EL&U member GreenRay points out.
from: French 'juste', meaning justly/fairly/righteously
(OED) (Etymology Online - http://tinyurl.com/etymonline)
from: Old French 'juste' , meaning as above
(Wiktionary - https://en.wiktionary.org/wiki/juste)
from: Latin 'justus', an alternative version of the Latin 'iustus'
(Wiktionary - (1) https://en.wiktionary.org/wiki/juste - (2) https://en.wiktionary.org/wiki/justus#Latin)
otherwise: Latin 'iustus' meaning just, righteous, lawful, legal, exact, straight direct.
(Wiktionary - https://en.wiktionary.org/wiki/iustus#Latin)
from: proto-Italic 'jowestus' meaning just, lawful
(Wiktionary - http://tinyurl.com/jowestus)
from: proto Italic 'jowos' meaning oath, law
(Wicktionary - http://tinyurl.com/jowos)
Note that the 'closely or proximate' meaning now included in the many senses of the word 'just' isn't reflected in the origins of the word, and seems to be quite at odds with the sense of 'exact' that is implied by the Latin root 'iustus'. The OED simply notes that the adverbial use of the French 'juste' included such meanings as 'exactly', 'precisely', 'verily', 'actually', and 'closely'. The meaning, closely seems somewhat at odds with the other senses of the French adverb. The question might be asked, did something happen to the word during its development in French?
It is quite feasible that a simple drift in the use of language accounted for the broadening of the sense of the word 'just' to incorporate 'close to' or 'almost'. However, the OED's entry for the word 'adjust' makes an interesting observation (noting that we will be coming back to the word 'just' after this small digression):
adjust adjust, v.
[a. 16th c. Fr. adjuste-r (now ajuster). The OFr. ajuster, ajoster, ajouster = It. aggiustare, aggiostare (:—late L. adjuxtāre), gave rise to a med.L. adjustare, which was naturally, though erroneously, taken as a derivative of ad + jūstus, and so consciously used. After Fr. ajouster became ajouter, so that its formal relationship to aggiustare and adjustare was lost sight of, a new Fr. adjuster was formed after the latter, and received those senses of ajouster, which seemed to approach to L. jūstus, Fr. juste. In It. and Sp. also the mod. sense of aggiustare, ajustar, has been influenced by association with jūstus. Thus mod.Fr. ajuster may be viewed as a refashioning or re-forming of OFr. ajouster:—adjuxtāre, after à+ juste. See also adjoust.]
So, according to the OED, at some point (16th century?) the French conflated the senses of the Latin word 'adjuxtāre' and the Latin word 'justus'. So what is 'adjuxtāre'? According to Wiktionary (http://tinyurl.com/adjuxtare) was derived from:
Latin adjuxtare, from Latin ad + juxta near to, nigh. Compare adjutage, adjustage, adjust. ajutage
If we follow the Latin 'juxta' we start to get meanings very similar to some of those non-exact senses of the French adverb 'juste' (http://tinyurl.com/nvbkk3d):
juxtā , adv. and prep. sup. form from jugis; v. jungo,
I.near to, nigh.
A. Lit., of place.
1. Of several objects, near together, in close proximity
2. In gen., near at hand, near, near by, hard by, close to, by the side of
B. Transf., in like manner, equally, alike
9: just as if , = non secus ac si, Cic. post Red. ap. Sen. 8, 20:
II. Prep., with acc., very near, close to, near to, hard by
A. Lit., of place
1. Next to, immediately after, beside, on a par with
2. Near, approaching to, like, almost the same as
So the interesting speculation is whether the French might have roughly handled the word 'juste' in the same way they misattributed and re-fashioned their understanding of the word 'adjust' . Could the Latin meaning of 'juxta' (near to) have leaked into the French word 'juste' in the same way the the sense of 'justus' (exact, righteous) had leaked into the French word 'adjust' which was properly derived from the Latin adjuxtāre (near to) as attested by the OED?
We might thicken the soup by mentioning at this point that the English word 'joust' used to be written as 'just' and spoken to rhyme with the word 'must'. 'Joust' is derived from the Latin 'juxtā', as the OED has it:
joust, just, v.
(dʒaʊst, formerly dʒuːst, dʒʌst)
Forms: 3–7 iust(e, 4–7 ioust(e, (5 youst, yust, iowst, iost, 6 iuyst), 7– just, joust.
[a. OF. juste-r (11th c.), joster (12th c.), jouster (13th c.) = Pr. justar, jostar, Sp., Pg. justar, It. giostrare:—late pop.L. juxtāre to approach, come together, meet, f. juxtā near together. The sense ‘approach, join’, remained in OF.
The historical Eng. spelling from the 13th c. is just: cf. the cognate adjust:—L. adjuxtāre. Under later French influence, joust was used sometimes by Gower, Caxton, Spenser, and Milton, was preferred by Johnson, and used by Scott, and is now more frequent; but the pronunciation remained as in the historical spelling; the pronunciation (dʒaʊst) (formerly dʒuːst) is recent, and suggested by the spelling joust.]
So to sum it up, I believe that GreenRay is correct in the sense that most of the meanings can be summarised as 'exactly or only', or 'closely or proximately in space, quantity or time'. But the contradiction between these two senses, as Rusty Tuba has pointed out, is difficult to put aside without comment. How can the word 'just' mean 'exactly', but at the same time also being used to mean 'almost' or 'proximate'?
However, if words derived from the Latin 'justus' (exact) and from 'juxto' (proximate) have over the ages shared spellings and pronounciation - as indicated above - then there may have been even more instances of erroneous derivation and attribution and 'meanings influenced by association' than the OED has so far mentioned. Perhaps that massively polysemous (thank you John Lawler) word 'just' has also drawn some of its meaning from the Latin 'juxtā', in addition to the meaning it inherited from its proper root, the Latin 'iusta'.