HYDRA, Greece—As Greece’s bailout talks with creditors enter their final days, many of its people are caught between relief that a breakthrough seems near and anxiety that it could all still go wrong.

On this picturesque island south of Athens, residents have in recent days debated whether a bailout deal will come through—and how to handle possible capital controls and bank closures if the talks in Brussels break down.

“Older people with pensions are worried,” said Pete Kalamatianakis, who operates a grocery store on the waterfront and says he chats to about 300 people daily. He has heard rumors that food could become scarce if cargo ships stop arriving on the island.

“Nothing’s going to happen,” said Dimitra Kavoukopoulou, proprietor of the women’s clothing store Dimitra next door to the grocery. The fears of some of the island’s residents are unwarranted, she said: “They will make a deal no matter what.”

Greece’s government is trying to win the approval of European institutions and the International Monetary Fund for a package of austerity measures and other economic policies in time for meetings of eurozone finance ministers on Wednesday and European leaders on Thursday. A deal would—for the time being—end months of tension between the government in Athens and its creditors over how much more austerity is needed to repair Greece’s public finances.

A deal this week would unlock bailout financing just in time for Greece to repay IMF loans that fall due on June 30. Failure to agree to terms in Brussels would very likely lead to the withdrawal of central-bank support for Greece’s banks—forcing Greece to impose capital controls that restrict ATM withdrawals and many financial transactions.

Caroline Waller, a Frenchwoman who has lived in Hydra for 10 years, said that she is monitoring the Brussels talks closely as she weighs whether to stop taking credit cards at her shop, which sells leather goods, jewelry and other Greek-made arts and crafts.

“It takes two days for the money to get into my account when I accept cards,” she said. “What if the banks close meanwhile? I am taking a risk.” But, she adds, “It is better for me to do business than to say ‘cash only’ and lose customers.”

She said that even if the current drama passed, she was concerned that the crisis could flare up again—in months, or years. “I don’t know if I will be able to stay in Greece,” she said.

In Athens, meanwhile, rival demonstrations have in recent days taken turns at occupying the city’s main plaza, Syntagma Square, protesting alternately for a deal with Europe and against the austerity measures that would be attached to a deal.

On Monday night, as eurozone leaders met in Brussels, it was the turn of supporters of an agreement with Europe, and of Greece remaining in the euro, to gather on Syntagma. The demonstration drew a few thousand people, including many supporters of centrist or conservative opposition parties. European Union flags, bearing gold stars on a blue background, mingled with Greek flags among the crowd. Those attending included many well-heeled denizens of Athens who, by their own account, rarely if ever join demonstrations.

Olga Alexopoulou, a 62-year-old pensioner in Tory Burch pumps, said she was losing patience with the government as she waved small Greek and EU flags in her left hand. “We think our future is in Europe…I’m very worried,” she said.

The square, in front of the Greek Parliament, has been the theater of vicious battles between antiausterity protesters and riot police for the past few years. But Tuesday’s rally was calm and orderly, with sporadic chants of “Greece! Europe! Democracy!”

The protest was briefly disrupted when around a dozen hooded protesters, mostly young men, threw a few Coca-Cola bottles at the pro-euro rally, causing the riot police to intervene. Within 20 minutes, most had been detained by police, and the pro-euro protesters screamed “Fascists! Sloths!” at them as they were taken away in police vehicles.

Some conservative members of parliament, among them two former ministers in the previous government, observed the scene from the western gate of the Parliament building and shook the hands of departing protesters.

“People are really stressing now, they’re reaching their limits,” said Giorgos Koumoutsakos, a lawmaker with the conservative New Democracy party.

Inside the cafe in the Parliament building on Tuesday, where lawmakers and their aides gather to smoke and swap gossip, members of the ruling left-wing Syriza party said they were unsure what kind of a deal their prime minister was negotiating in Brussels.

Speaking privately, some said they would have to see the final deal before deciding how to vote. Others indicated they would wait to see what the party hierarchy decided. Several Syriza lawmakers, however, have already threatened to reject the expected deal, on the grounds that it violates Syriza’s election promise to end austerity.

In the corridors of Parliament, the big unknown is how many will carry out that threat.

“If the government doesn’t have the trust of its [members of Parliament,] it cannot operate and it goes to elections,” government spokesman Gabriel Sakellarides told Greek television station Antenna on Tuesday.