It's important to review the wider context and ask the basic questions.
1 Corinthians 9:1-18 (DRB)
1 Am not I free? Am not I an apostle? Have not I seen Christ Jesus our Lord? Are not you my work in the Lord? 2 And if unto others I be not an apostle, but yet to you I am. For you are the seal of my apostleship in the Lord. 3 My defence with them that do examine me is this. 4 Have not we power to eat and to drink? 5 Have we not power to carry about a woman, a sister, as well as the rest of the apostles, and the brethren of the Lord, and Cephas? 6 Or I only and Barnabas, have not we power to do this? 7 Who serveth as a soldier at any time, at his own charges? Who planteth a vineyard, and eateth not of the fruit thereof? Who feedeth the flock, and eateth not of the milk of the flock? 8 Speak I these things according to man? Or doth not the law also say these things? 9 For it is written in the law of Moses: Thou shalt not muzzle the mouth of the ox that treadeth out the corn. Doth God take care for oxen? 10 Or doth he say this indeed for our sakes? For these things are written for our sakes: that he that plougheth, should plough in hope; and he that thrasheth, in hope to receive fruit. 11 If we have sown unto you spiritual things, is it a great matter if we reap your carnal things? 12 If others be partakers of this power over you, why not we rather? Nevertheless, we have not used this power: but we bear all things, lest we should give any hindrance to the gospel of Christ. 13 Know you not, that they who work in the holy place, eat the things that are of the holy place; and they that serve the altar, partake with the altar? 14 So also the Lord ordained that they who preach the gospel, should live by the gospel. 15 But I have used none of these things. Neither have I written these things, that they should be so done unto me: for it is good for me to die, rather than that any man should make my glory void. 16 For if I preach the gospel, it is no glory to me, for a necessity lieth upon me: for woe is unto me if I preach not the gospel. 17 For if I do this thing willingly, I have a reward: but if against my will, a dispensation is committed to me: 18 What is my reward then? That preaching the gospel, I may deliver the gospel without charge, that I abuse not my power in the gospel.
What's going on here? St. Paul is relinquishing rights he has as a believer in Christ; and more specifically in this argument, as a pastor of the church.
This is done voluntarily for the given reason that there is a chance (however small) that making use of these legitimate things could cause scandal to others—causing "a weak brother to perish, for whom Christ died" (by being perceived in anyway as scandalous, perhaps).
What exactly is it he is giving up, though?
Eating and Drinking
Clearly the converse is not starvation; no one is exepected to undertake such a mortification as a pastor of the church. In context, he just got done telling us (1 Cor 8) he would abstain from certain foods indefinitely, if it meant not scandalizing others who were sensitive to such (this doesn't refer to veganism or anything of the sort, but religious aversion to certain foods).
His main point with this issue of foods was about how abstaining from certain foods for this reason wasn't to imply doctrinal objection to the foods intrinsically, but that if it can be avoided and also prevents others who would be scandalized by it, then St. Paul and anyone else who sets such limits on themselves should not be challenged on it.
So in context, then, "have we not the right to eat and to drink?" means 'have we not the right to eat or drink what we want?'
The Greek words translated "a believing wife" in the translation you provided are αδελφην γυναικα (adelphaen gunaika). The first word is αδελφην (sister) and the second γυναικα (woman). Those are their plain meaning without interpretative lenses. So without interpretation or context, we simply have "sister woman" ('sister wife' is equally meaningless in English).
In English we have heard people refer to their husband or wife as 'my man' or 'my woman' respectively. Whereas this is considered to be a somewhat derogatory way of referring to one's spouse, this simply doesn't carry over to Koine Greek: the 'woman' of a certain sentence or context is just inferred to have the 'wife' meaning (especially when accompanied with the possessive, e.g., my woman/wife) (Cf. Mt 22:28 vs. Jn 4:7). The same goes for the Greek man/husband: ανηρ (anaer) (Cf. Jn 4:18). Considered somewhat derogatory in English in certain contexts, the Greek word is simply multivalent, and has a dual function: a 'woman' in a certain context simply means the 'wife' of the situation (especially when with the possessive pronoun of course).
My understanding of the "sister woman" they have the right to carry along with them is that "sister" is (given our larger context mentioned before) in contradistinction to an in any way, shape or form, sexualized female. She is considered a sister. Such would be legitimate and allowed St. Paul, his argument goes. But he has opted, chosen freely, to forego the whole thing, just to avoid even potential scandal to others. Maybe Paul is under heavier criticism than the other Apostles, given how he came to be one. Maybe that's why he has to remove all obstacles to an un"hindered" preaching of the gospel.
My humble translation suggestion for this would be: "Do we not have the right to take a [believing] woman along with us [as] a sister?"
Or I only and Barnabas, have not we power to do this?
[This is a slightly archaic translation. Read: Or is it only Barnabas and I that have not the right to such?]
His point is that he and Barnabas are not foregoing these because they are somehow intrinsically bad in and of themselves. This is why he asks the rhetorical question, 'Don't we have the right?' To which the answer is, 'of course you do.' His argument is that they aren't refraining from these things because they have some moral aversion or doctrinal issue with them, but for other voluntary reasons particular to their evangelism (which as Paul notes in verses 8, 11, and 20 of chapter 9, takes the form of avoiding scandal regarding the New Testament lift on the prohibition of foods under pain of sin. "Know you not, that they who work in the holy place, eat the things that are of the holy place; and they that serve the altar, partake with the altar? 14 So also the Lord ordained that they who preach the gospel, should live by the gospel. 15 But I have used none of these things." They have the right, but are not using it for the reasons given.
The 'only' in 'Barnabas and I only' seems to be intended to hint at the absurdity of an inconsistent Christianity between the Apostles: one for Peter and the other Apostles, and then a different one with different laws for Paul and Barnabas, to the effect that some Apostles are allowed to eat certain things, whereas it's sin for others.