Exceptions have a significant performance impact, and making code exception safe distorts your interfaces. Insisting on no exceptions is reasonable for these and other reasons, even for new code.
In the C world this is extremely arduous since need to bubble the allocation-failure down through every level of code, adding cleanup to just about every function call. If you're using C++ and RAII you can just catch std::exception in one place to cover a huge number of places that an allocation can fail.
In modern systems I'm familiar with, malloc only reports failure on bogus inputs like -1, or address space exhaustion. Your process is likely to be killed before exhausting your address space (think iOS OOM handling, or Linux overcommit), especially on 64 bit. So checking for allocation success just isn't that useful any more.
The first is process limits.
The second is exhausting kernel data structure space for things like page mappings. I've seen this recently in AIX when allocating lots of memory that alternates mprotect permissions.
Gotta love putting forth every effort in software to keep the BOM down :)
Even in the absence of the OOM killer (i.e. the old days) you had to do this -- otherwise the machine might be swapping itself unresponsive for ages before you ever get malloc()==NULL
Are you talking about enabling exceptions, or throwing them?
I don't think it's very popular today. Modern implementation don't use this anymore and do not incur run-time overhead when no exception are thrown. However, they do incur some space overhead: the implementation need to maintain pretty large tables to know what needs to be unwound (which destructors to call, in which order) when an exception is thrown. In most environment this space overhead shouldn't be a problem (PC, server). But for embedded development it may be a problem. And then disabling exception to save space brings back the issues with error checking (or the lack of it...).
For more details, search for C++ exception implementation and you should find all the info you need. Or look into G++ documentation for example, this topic is covered somewhere.