Is it possible to add two signed 8-bit numbers together and set both the carry and overflow bits?
You don't have access to the flags in C, even if you could get the compiler to generate code that set them, you have have no way to use them.
You can write your own add routine in C that will return carry and overflow flags for signed 8-bit operands. If you're referring to the hardware carry and overflow bits inside the processor, no, that cannot be done portably in C.
Per your comments, your question seems to be "is it possible to have both carry and overflow set for a two's complement add involving signed number?" It is. The typical implementation is to take the exclusive-OR of the carry-in for the last adder with the carry-out at the end of the chain -- hence, an overflowing addition of negative numbers will cause the carry-out bit to be set and the overflow bit to be set.
Here's an example, add -1 to -128:
Carry 10000 0000
1000 0000 (-128)
1111 1111 (-1)
---------
0111 1111 (oops, this is 127!)
Carry will be set, since the last add resulted in a carry -- and overflow will be set based on the rule above (also, note that -128 added to -1 is obviously not 127)