Exceptions are possible in C. See the documentation for setjmp() and longjmp().
That said, exceptions are just "kicking the can down the road" for error handling. If a function call can fail, then you should check the return code. If you don't want to write with proper error reporting/recovery code immediately, there is always the assert() macro, e.g.:
if(func_which_might_fail() == ERROR_OCCURRED) assert(0);
If assert(0) gets called the program will stop immediately, and you can inspect the problem in detail with a debugger. Easy peasy.
Urgh, no. Never, ever do this. assert() usually becomes a no-op when compiling stuff without debug flags, so binaries shipped to the field will mysteriously ignore errors in ways that can't be reproduced in a debugger.
One thing that shouldn't change between debug and production code is error detection and handling. In this sense, exceptions are ideal, as handling only needs to happen as and when errors occur, and the common case of no errors incurs little (setjmp/longjmp) to no (exception tables) overhead.
Plus exceptions are less laborious to handle. You can have a single try/catch for a whole block of code, instead of testing and handling each and every function that could fail. If you write exception safe code, then that's all you need to do, and you can clean up failed operations in the exception handlers.