One option, for Greece to return to the drachma, would be "catastrophic", he said, and not just for Greece itself.
"It would be a disaster for everyone involved. It would be a disaster primarily for the Greek social economy, but it would also be the beginning of the end for the common currency project in Europe," he told the BBC. "Whatever some analysts are saying about firewalls, these firewalls won't last long once you put and infuse into people's minds, into investors' minds, that the eurozone is not indivisible," he added.
Both ministers' words followed a declaration from Greek Prime Minister Alexis Tsipras that bargaining with Greece's creditors would soon end.
"We are on the final stretch of a painful and tough period shaped by the Government's negotiations with the institutions. Rest assured that in this negotiation we will not accept humiliating terms," Tsipras said. Most Greeks "want a solution and not just an agreement ... they support the Government in this tough negotiation", he said.
Varoufakis said using the common currency was now like using a "foreign currency", and any exit from the eurozone would be "a disaster".
"Trying to get out of it is tantamount to announcing a devaluation 10 months in advance."
Economists say if Greece were to leave the euro area, it could trigger huge levels of capital flight.
In turn, Greece would almost certainly have to resort to capital controls to stem the tide of money out of its domestic economy.
Ratings agency Moody's has warned there is now a "high likelihood" of such controls, which might be necessary to keep the Greek financial system alive. An estimated €30 billion has been withdrawn from Greek banks since snap elections were called last December.
Varoufakis said that at some point the Greek Government would have to choose between paying salaries and paying international creditors. The decision was one "no Minister of Finance should ever have to make".
The Government didn't need a referendum to refresh its mandate, he said, as was suggested by the German Finance Minister.