We are compiling an embedded C++ application that is deployed in a shielded device in an environment bombarded with ionizing radiation. We are using GCC and cross-compiling
What you ask is quite complex topic - not easily answerable. Other answers are ok, but they covered just a small part of all the things you need to do.
As seen in comments, it is not possible to fix hardware problems 100%, however it is possible with high probabily to reduce or catch them using various techniques.
If I was you, I would create the software of the highest Safety integrity level level (SIL-4). Get the IEC 61513 document (for the nuclear industry) and follow it.