On Deep Space Nine people commonly went to a place called the replimat to eat. With Bajor not yet a member of the Federation they didn't have the Federation's moneyless economy (however that's supposed to work). Did patrons have to pay for food and drink there (I'm going partly on the assumption the replimat was based on the automat places that used to exist on Earth IRL)?
The economy of Deep Space Nine is something of an enigma, compounded by its status as a de facto free port even as it's also serving as a regional wartime fortress. The Federation, of course, has not merely a cashless but a non-monetary economy (which, as you point out, is never really explained). At the same time, there are vendors on the station -- most notably Quark, but also a Klingon establishment -- who are only going to do business in hard currency, usually gold-pressed latinum, a substance that is implied to not be replicable (otherwise it would be worthless). Various cultures appear to use this hard-currency standard, in much the same way that many cultures used the Spanish silver 8 Real piece, which could be divided into eight bits.
On the flip-side, the station is playing continual host to members of various cultures who are not participants in the Federation's non-monetary system. That is, even if the station were truly a Federation Starbase, they would still have only cold, hard latinum to exchange for goods and services.
What remains ambiguous on-screen (probably deliberately) is the interface between Starfleet personnel and this hard-currency system. We sometimes see Starfleet officers handling latinum (particularly to play Quark's games, or, in Dax's case, to play games with Quark. She keeps winning at Tongo and seems to have quite a stash!), suggesting that there is some sort of exchange system, possibly unofficial. Quark also speaks occasionally of keeping tabs for various regulars, including Starfleet members. Turning it around, we have a few examples where non-Federation people (usually Quark) are expected to make some sort of recompense for damages done, or judicial fines under Bajoran law or station regulations.
Of course, you're asking about the replimat, and not Quark's, but unfortunately, the only establishment where we have any evidence, expressed or implied, is Quark's, and we're left to generalize and to some degree guess. We also know that there's a Klingon eatery, a Bajoran restaurant, and eventually Bolian and Vulcan establishments (Memory Alpha: Promenade). The Bolian and Vulcan establishments could certainly be expected to operate under Federation economic rules (whatever they are) for Starfleet members, but seem likely to take hard currency from others; and the Klingon restaurant also almost certainly deals only in hard currency, as the Klingons, even when allied with the Federation, do not seem to participate in their post-monetary economy.
Which brings us to the replimat, where all we can do is guess from the clues above. If the station truly is considered Bajoran sovereignty with Starfleet essentially acting as contractors (a status Sisko is frequently at pains to reinforce, particularly early in his tenure when he's trying, for example, to convince Quark to stay), then I would surmise that the replimat is also a hard-currency facility, and that the same exchange mechanisms--whatever they might be--which allow Starfleet officers to patronize other hard-currency facilities, apply there.
If, on the other hand, the replimat is a service being provided by Starfleet, then I would expect it to operate more under Federation economic rules, with the exchange working in reverse for people who wish to patronize it but don't participate in that system.
In the end, as fascinating as these details are, they never formed the focus of a story, and thus, all we're left with is a lot of hints from which to draw conjectures!