We simply don't handle exceptions thrown by containers, at least in application-level code.
I've been an engineer at Google Search working in C++ since 2008. We do use STL containers often. I cannot personally recall a single major failure or bug that was ever traced back to something like vector::push_back() or map::operator[] failing, where we said "oh man, we have to rewrite this code because the allocation could fail" or "dang, if only we used exceptions, this could have been avoided." Does a process ever run out of memory? Yes, but this is usually a simple mistake (e.g., someone added a large new data file to the program and forgot to increase the RAM allocation) or a catastrophic failure where there's no good way to recover and proceed. Our system already manages and restarts jobs automatically to be robust to machines with faulty disks, cosmic rays, etc., and this is really no different.
So as far as I can tell, there is no problem here.