The Athens stock exchange will also be closed as the government tries to manage the financial fallout of the disagreement with the European Union and the International Monetary Fund.
Greece's banks, kept afloat by emergency funding from the European Central Bank, are on the front line as Athens moves towards defaulting on a 1.6 billion euros payment due to the International Monetary Fund on Tuesday (local time).
Greece blamed the ECB, which had made it difficult for the banks to open because it froze the level of funding support rather than increasing it to cover a rise in withdrawals from worried depositors, for the moves.
Prime Minister Alexis Tsipras said the decision to reject Greece's request for a short extension of the bailout programme was "an unprecedented act" that called into question the ability of a country to decide an issue affecting its sovereign rights.
"This decision led the ECB today to limit the liquidity of Greek banks and forced the central bank of Greece to propose a bank holiday and a restriction on bank withdrawals," he said in a televised address.
Amid drama in Greece, where a clear majority of people want to remain inside the euro, the next few days present a major challenge to the integrity of the 16-year-old euro zone currency bloc. The consequences for markets and the wider financial system are unclear.
Greece's left-wing Syriza government had for months been negotiating a deal to release funding in time for its IMF payment. Then suddenly, in the early hours of Saturday, Tspiras asked for extra time to enable Greeks to vote in a referendum on the terms of the deal.
Creditors turned down this request, leaving little option for Greece but to default, piling further pressure on the country's banking system.
The creditors want Greece to cut pensions and raise taxes in ways that Tsipras has long argued would deepen one of the worst economic crises of modern times in a country where a quarter of the workforce is already unemployed.
Pro-European Greek opposition parties have united in condemning the decision to call the referendum on the bailout terms, but many people are supportive.
"I want him (Tsipras) to knock his fist on the table and to say 'enough!'," said Athens resident Evgenoula.
Many leading economists have voiced sympathy with the Greek government's argument that further cuts in spending risk choking off the growth which would give Greece some prospect of servicing debts worth nearly twice its annual national income.
The IMF has pressed European governments to ease Athens' debt burden, something most say they will only do when Greece first shows it is trimming its budget.
Stocking up on cash
Long lines formed outside many ATMs on Sunday, including some of 40 to 50 people outside some in central Athens.
The Bank of Greece said it was making "huge efforts" to ensure the machines remained stocked.
The German foreign ministry said tourists heading to Greece should take plenty of cash to avoid possible problems with local banks and some tourists said they were joining the ATM queues.
"I am trying to go over to the bigger banks," said Cassandra Preston, a Canadian tourist. "I am here for another month and I would like to make sure I have some cash on me."
The ECB has kept the banks afloat in recent days with increases in its funding line, a form of overdraft with the euro zone's central bank system.
But on Sunday it said it would hold the funding line at the same level as Friday, despite the deposit outflows. The central bank said it was monitoring the situation and stood ready "to reconsider its decision."
There is growing opposition to the funding line because it would fall to the bloc's other members to pay if Greece were to leave the euro zone.
In economic powerhouse Germany, other southern states that have suffered austerity in return for EU cash and poor eastern countries with living standards much lower than Greece's, many voters and politicians have run out of patience.
German Finance Minister Wolfgang Schaeuble openly questioned the solvency of Greek banks - a key condition to qualify to receive such finance.
"The ECB has always said that as long as Greek banks are solvent, then emergency loans, the ELA, can be granted," he said on Saturday. "And now there is naturally a new situation that because of the developments the liquidity and solvency of Greek banks, or some Greek banks, could be in doubt."
German Chancellor Angela Merkel has invited leaders of all the major German parties to a meeting in Berlin on Monday to discuss the crisis.
Deal still possible
The 18 other countries sharing the euro countries have blamed Greece for breaking off negotiations and pledged to do whatever it takes to stabilize the common currency area.
European Council President Donald Tusk said on Sunday he was in contact with all the governments of the euro zone to ensure Greece remained in the single currency
Several officials said there was still time to return to the negotiating table.
"To those who wonder what's next, 1. Greece should stay in euro; 2.The door is still open for negotiations on latest EU Commission proposals," EU Economics Commissioner Pierre Moscovici said.
French Prime Minister Manuel Valls on Sunday urged the Greeks to continue talks.
"I cannot resign myself to Greece leaving the euro zone ... We must find a solution," Valls told Europe 1, Le Monde and iTELE in a joint interview.
International Monetary Fund boss Christine Lagarde said that if the July 5 vote produced "a resounding yes" to remain in the euro and fix the Greek economy then the creditors would be willing to make an effort.
