Lexical Analysis
The verbs ἐκδημέω (ekdēmeō) and ἐνδημέω (endēmeō) are related to the noun δῆμος (dēmos), which has several meanings.
LSJ notes,1

Other related words include the adjectives ἔκδημος (ekdēmos) and ἔνδημος (endēmos),2 as well as the nouns ἐκδημία (ekdēmia) and ἐνδημία (endēmia).
The apostle Paul uses the verbs ἐκδημέω and ἐνδημέω according to δῆμος in the sense of “country.” Thus, the verb ἐκδημέω means “to be out of (ἐξ) the country,” which is synonymous with “to be away from home,” “to travel abroad,” or “to emmigrate.” The verb ἐνδημέω means to be “in (ἐν) the country,” which is synonymous with “to be at home.”
Secular Literature
2 Cor. 5 does appear to be the only content wherein both verbs occur in close proximity to one another.3 However, both verbs do occur often by themselves.
ἐνδημέω and related words
While writing about rulers living at home and those living abroad, Aristotle refers to the former by the adjective ἔνδημοι and the latter by ὑπερόριοι (“abroad").4 Hesiod contrasts «ξείνοισι καὶ ἐνδήμοισι», the former commonly being translated as “strangers” or “foreigners” (i.e., those who live abroad).5 Xenophon refers to possessions located «τῶν ἐνδήμων καὶ τῆς ὑπερορίας», “at home and abroad.”6
ἐκδημέω and related words
Euripides distinguishes between “loved ones abroad” (τῶν...ἐκδήμων φίλων) and “those at home” (τοὺς κατ᾽ οἶκον ὄντας).7
Exegesis
In 2 Cor. 5:6-9, the apostle Paul writes,
6 Therefore, always being confident and knowing that while we are at home in the body, we are away from the Lord, 7 for we walk by faith, not by sight, 8 but we are confident and well pleased rather to be away from the body and to be at home with the Lord. 9 For this reason, we labour to be accepted of him, whether we are at home or away.
Ϛʹ θαρροῦντες οὖν πάντοτε καὶ εἰδότες ὅτι ἐνδημοῦντες ἐν τῷ σώματι ἐκδημοῦμεν ἀπὸ8 τοῦ κυρίου Ζʹ διὰ πίστεως γὰρ περιπατοῦμεν οὐ διὰ εἴδους Ηʹ θαῤῥοῦμεν δὲ καὶ εὐδοκοῦμεν μᾶλλον ἐκδημῆσαι ἐκ τοῦ σώματος καὶ ἐνδημῆσαι πρὸς τὸν κύριον Θʹ διὸ καὶ φιλοτιμούμεθα εἴτε ἐνδημοῦντες εἴτε ἐκδημοῦντες εὐάρεστοι αὐτῷ εἶναι
Earlier in the same chapter, the apostle Paul distinguished between “our earthly house of [this] tabernacle" and “a building of God, a house made without hands, eternal in heaven.”9 While some suppose the latter refers to heaven, it is more likely referring to the incorruptible, immortal, spiritual body (σῶμα πνευματικόν),10 for the apostle Paul also writes,
2 For in this we groan, earnestly desiring to be clothed upon with our house that is from heaven.
Βʹ καὶ γὰρ ἐν τούτῳ στενάζομεν τὸ οἰκητήριον ἡμῶν τὸ ἐξ οὐρανοῦ ἐπενδύσασθαι ἐπιποθοῦντες
“In this” refers to the “earthly house of this tabernacle.” The apostle Paul says that we groan in such a tabernacle (i.e., mortal body), desiring to be invested with or clothed upon (ἐπενδύσασθαι) by “our house that is from heaven.” He further states that if we are clothed upon with the latter house, “we shall not be found naked,” that is, as an incorporal soul/spirit, destitute of a body.11 Hence, the house from heaven that we are to be clothed upon or invested with is the incorruptible, immortal, spiritual body.
To the verses in question: 2 Cor. 5:6-8. In v. 6-7, the apostle Paul says, “Therefore, while we are confident and knowing that while we are at home in the body, we are away from the Lord --- for we walk by faith, not by sight.” When he says “while we are at home in the body,” what body is he referring to? It seems to me to refer to “our earthly house of [this] tabernacle.”
Now, if that same body is dissolved (cp. v. 1), then we are clothed upon our house that is in heaven, the incorruptible body. This clothing occurs upon the resurrection from the dead, upon the advent of the Lord Jesus Christ. Once the saints are changed from mortal to immortal, from the physical or soulish body to the spiritual body, “thus they shall always be with the Lord.”12
I understand many interpret 2 Cor. 5:8 as though it is referring to going to heaven as an incorporal soul upon one’s death, and so I hesitate to interpret that verse as if to contradict such a notion, but it seems to me that the means by which the apostle Paul is “willing rather to be away from the body and to be at home with the Lord” is via his resurrection and change to an immortal body. In that manner, he is with the Lord (in heaven) ► with a body. That is not to say that there is no such thing as an intermediate state, or a soul, for I certainly believe there is. I simply don’t see that the apostle Paul was endeavoring to make that point. Rather, he seems intent on discussing the mortal (corruptible, physical) v. immortal (incorruptible, spiritual) bodies.
Footnotes
1 p. 386
2 From which the English adjective “endemic” is derived.
3 Plato comes quite close to using both together. See Laws, Book 9, §864e: «οὕτω δ᾽ εἰς ἄλλην χώραν καὶ τόπον ἀπελθὼν οἰκείτω τὸν ἐνιαυτὸν ἐκδημῶν, πρότερον δὲ ἐλθὼν τοῦ χρόνου ὃν ὁ νόμος ὥρισεν, ἢ καὶ πάσης ἐπιβὰς τῆς οἰκείας χώρας, ἐν δημοσίῳ δεσμῷ δεθεὶς ὑπὸ τῶν νομοφυλάκων δύο ἐνιαυτούς, οὕτως ἀπαλλαττέσθω τῶν δεσμῶν.»
4 Aristotle. Athenian Constitution. Ch. 24.
5 Hesiod. Works and Days. §225.
6 Xenophon. Anabasis. Book 7, Ch. 1, §27.
7 Eurpides. Andromache. §1051.
8 It is interesting that the apostle Paul used the preposition ἀπὸ rather than ἐκ. ἐνδημέω is followed by ἐν, and ἐκδημέω should be followed by ἐκ (it is in v. 7: «ἐκδημῆσαι ἐκ τοῦ σώματος»). This seems to indicate that ἐκδημέω and ἀποδημέω are synonymous. As expected, Maltby (below) does note them to be synonyms (p. 199):

9 2 Cor. 5:1: «ἡ ἐπίγειος ἡμῶν οἰκία τοῦ σκήνους» versus «οἰκοδομὴν ἐκ θεοῦ ἔχομεν οἰκίαν ἀχειροποίητον αἰώνιον ἐν τοῖς οὐρανοῖς»
10 1 Cor. 15:44
11 That is, the apostle Paul either hope or expected to be changed from mortal to immortal without suffering death. cp. 1 Cor. 15:50-56
12 1 Thes. 4:17: «καὶ οὕτως πάντοτε σὺν κυρίῳ ἐσόμεθα»
References
Liddell, Henry George; Scott, Robert; et al. A Greek-English Lexicon. 9th ed. Oxford: Clarendon, 1940.
Maltby, Edward. A New and Complete Greek Gradus, Or, Poetical lexicon of the Greek Language. London: Cadell, 1830.