Let’s say a Eurozone country faces a bank failure and the debt of the failing bank is very large relative to that country’s gdp. This could happen in many countries. The fiscal authority of that country cannot do the bailout on its own, in part because the resources are not there and in part because the country lacks the political unity for raising taxes. The other EU countries cannot be persuaded to ante up. The country in question either loses its major bank, and suffers the concomitant fall out, or it creates "on the spot" monetary policy to save the bank. That means creating a domestic currency and suddenly announcing that the accounts of the bank will be reimbursed in terms of that currency rather than Euros.
I believe the odds of this outcome are relatively small.
That said, the "easy" option is for the ECB to do the bailout in terms of Euros. A non-unanimity-requiring procedure for this should be worked out in advance to a greater degree than is currently the case. And such a procedure may need to be worked out soon, while it is still unclear who would be the winners and losers from such an arrangement.