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
This answer assumes you are concerned with having a system that works correctly, over and above having a system that is minimum cost or fast; most people playing with radioactive things value correctness / safety over speed / cost
Several people have suggested hardware changes you can make (fine - there's lots of good stuff here in answers already and I don't intend repeating all of it), and others have suggested redundancy (great in principle), but I don't think anyone has suggested how that redundancy might work in practice. How do you fail over? How do you know when something has 'gone wrong'? Many technologies work on the basis everything will work, and failure is thus a tricky thing to deal with. However, some distributed computing technologies designed for scale expect failure (after all with enough scale, failure of one node of many is inevitable with any MTBF for a single node); you can harness this for your environment.
Here are some ideas:
Ensure that your entire hardware is replicated n times (where n is greater than 2, and preferably odd), and that each hardware element can communicate with each other hardware element. Ethernet is one obvious way to do that, but there are many other far simpler routes that would give better protection (e.g. CAN). Minimise common components (even power supplies). This may mean sampling ADC inputs in multiple places for instance.
Ensure your application state is in a single place, e.g. in a finite state machine. This can be entirely RAM based, though does not preclude stable storage. It will thus be stored in several place.
Adopt a quorum protocol for changes of state. See RAFT for example. As you are working in C++, there are well known libraries for this. Changes to the FSM would only get made when a majority of nodes agree. Use a known good library for the protocol stack and the quorum protocol rather than rolling one yourself, or all your good work on redundancy will be wasted when the quorum protocol hangs up.
Ensure you checksum (e.g. CRC/SHA) your FSM, and store the CRC/SHA in the FSM itself (as well as transmitting in the message, and checksumming the messages themselves). Get the nodes to check their FSM regularly against these checksum, checksum incoming messages, and check their checksum matches the checksum of the quorum.
Build as many other internal checks into your system as possible, making nodes that detect their own failure reboot (this is better than carrying on half working provided you have enough nodes). Attempt to let them cleanly remove themselves from the quorum during rebooting in case they don't come up again. On reboot have them checksum the software image (and anything else they load) and do a full RAM test before reintroducing themselves to the quorum.
Use hardware to support you, but do so carefully. You can get ECC RAM, for instance, and regularly read/write through it to correct ECC errors (and panic if the error is uncorrectable). However (from memory) static RAM is far more tolerant of ionizing radiation than DRAM is in the first place, so it may be better to use static DRAM instead. See the first point under 'things I would not do' as well.
Let's say you have an 1% chance of failure of any given node within one day, and let's pretend you can make failures entirely independent. With 5 nodes, you'll need three to fail within one day, which is a .00001% chance. With more, well, you get the idea.
Things I would not do:
Underestimate the value of not having the problem to start off with. Unless weight is a concern, a large block of metal around your device is going to be a far cheaper and more reliable solution than a team of programmers can come up with. Ditto optical coupling of inputs of EMI is an issue, etc. Whatever, attempt when sourcing your components to source those rated best against ionizing radiation.
Roll your own algorithms. People have done this stuff before. Use their work. Fault tolerance and distributed algorithms are hard. Use other people's work where possible.
Use complicated compiler settings in the naive hope you detect more failures. If you are lucky, you may detect more failures. More likely, you will use a code-path within the compiler which has been less tested, particularly if you rolled it yourself.
Use techniques which are untested in your environment. Most people writing high availability software have to simulate failure modes to check their HA works correctly, and miss many failure modes as a result. You are in the 'fortunate' position of having frequent failures on demand. So test each technique, and ensure its application actual improves MTBF by an amount that exceeds the complexity to introduce it (with complexity comes bugs). Especially apply this to my advice re quorum algorithms etc.