The Greek people are in similar dire straits as they prepare to vote on a future in which they face two painful prospects: the slow grind of years more of austerity cuts or the country's potentially catastrophic exit from the euro.
The question is whether their vote on Sunday can help them escape either. "Yes" to more budget cuts in exchange for a financial aid package for the country? Or reject it in the hope it will not lead the country out of the euro?
The referendum question makes no reference to Greece's future in the currency union. It is on a set of proposals that European creditors say they have withdrawn following the failure to forge a deal with Greece before an end-June deadline.
For the radical left-led Greek government, the proposals were unacceptable. It's urging a "no" vote and says that will have no impact on Greece's euro status.
Proponents of a "yes" vote, including a parade of former prime ministers and the main opposition party, say backing the government will jeopardize Greece's place in the euro. Instead, they argue that by voting "yes" Greece would get a new deal quickly to shore up the economy.
In fact, what might happen in each case is unclear. Analysts in the world's biggest investment banks are putting percentage probabilities to outcomes, such as Greece's exit from the euro, but no one knows for sure.
Here's a look at the events that each vote might trigger.
IF THE PEOPLE SAY 'NAI'
A "yes" vote backing the reform proposals creditors had made would likely see Greece turn immediately to talks on a new rescue package. Whether that leads to a swift deal that might allow Greece to reopen its banks and restore a semblance of normality to the life of citizens and tourists is another question.
Much would likely depend on what happens on the political front.
The government has said it will respect the verdict.
Greek Finance Minister Yanis Varoufakis has said he'll resign in the event of a "yes" vote and Prime Minister Alexis Tsipras has hinted as much. If the government does not collapse, it could try to build a new coalition with other parties, Varoufakish hinted.
It's not clear, however, if that would involve new elections. That would take time and without financial assistance, Greece would surely go bankrupt.
Greece is no longer in a bailout program since its previous package expired Tuesday. So it would have to negotiate a new one with its creditors that involves more money for the government and the banks and new economic measures.
That is unlikely to be agreed on overnight, meaning the harsh controls on money withdrawals and transfers may remain in place for longer than anticipated. The Greek government had to put those limits when a run on the banks started last weekend and the European Central Bank refused to increase the emergency credit it allows the banks to draw on.
Varoufakis says banks will reopen Tuesday whatever the referendum's outcome. That's unlikely to happen unless the ECB agrees to increase the credit to Greek banks. And the ECB would be under huge pressure not to do so until Greece has a new, comprehensive financial rescue package.
An additional difficulty is that Greece's creditors are singing from different hymn sheets. The International Monetary Fund has said it will not get involved in a third bailout unless it includes substantive debt relief for Greece. The Europeans, on the other hand, have ruled out debt relief until Greece makes its reforms.
"A new agreement will likely take time and the ultimate outcome may require even greater fiscal and structural commitments than the existing proposal," said George Saravelos, a strategist at Deutsche Bank. "The extent to which there is a sufficient political shift in Greece to allow this to materialize remains the key source of uncertainty."
IF THEY SAY 'OXI'
Despite the Greek government's assertion that a "no" vote will not lead to a euro exit, most people agree it would open up more uncertain outcomes, especially if the ECB calls time on the life-support measures to Greece's banks.
A number of European politicians, including Jeroen Dijsselbloem, the top eurozone official, have said a "no" vote would jeopardize Greece's place in the euro.
Others, such as the leaders of France and Italy, appear to be holding the door ajar for further talks. Even Wolfgang Schaeuble, the tough-talking German finance minister, has said the country could stay in the euro in the event of a "no" vote.
But investors are likely to be worried in case of a "no" vote amid fears it increases the chance of a Greek exit from the euro, or Grexit. Markets will open first in Asia.
The word 'Grexit' has dominated the past months of negotiations on Greece. But the country will not return to the drachma as soon as Monday. Rather, the risk increases the longer there is a deadlock in talks. Without a deal and without money, Greece will default on more of its debt repayments and will not be able to afford the day-to-day spending on salaries and pensions. The banks will run dry, even with the cash withdrawal limits.
In such a case, printing a new currency may be the only option available, which almost everyone thinks will be a short-term disaster for the Greek economy.
"A 'no' will result in Grexit with an uncertain future and high costs to Greek society, at least initially," said Guntram Wolff, director of think-tank Bruegel.
Odysseus got through his ordeal, but the monster ate six of his men — not a great thought for Greeks seeking to navigate the straits of bankruptcy and national pride.