However, you can get very far with the built-in real-time capabilities. If your software stack can deal with occasional jitter in sensor updates (and normally you have to take that into account), you can compile a stock kernel with PREEMPT_RT and can use C to build software that has to do IO every 1ms. In POSIX. That is huge.

What would be a real-time OS? I'd be interested in hooking up a bunch of independent boards that could interact with each other, and have as quick as a response to sensors as possible... Do you have any advice as to where to start with this type of project?

Sensor latency will be measured in milliseconds in Linux. A real time OS is more necessary when you have feedback loops, inverted pendelum problems, bitbanging protocols, etc. Not being a real time OS just means that when you say "ping me in X time" it might ping you in X +- Y. Y tends to be too big for critical applications, but generally small enough that it won't matter.