We all know what a grave is. If you don't, then it's a hole dug in the ground to bury dead people. What not as many people understand is that 'grave' has a second meaning. It also means serious.
We tend to see the this definition when we talk about 'gravity'. Inevitably, it is "the gravity of the situation." And it's usually about people not understanding said gravity. But the point remains, 'grave' means 'serious'.
I find 'grave' to be an interesting word in this context because it is difficult to use. People who intentionally use the word are either setting up for some pun (e.g. The zombie apocalypse was a grave situation) or trying to sound educated and sophisticated by using a somewhat archaic definition of a common word (or using a less common word in place of a simpler one, like 'serious').
The only thing I have that comes close to being a rule for this is, "Do what feels right." It's kind of a cop out, because that means I can't describe or explain it. I admit that. But I also say that the matter comes down to tone. It is a sound issue. Can the person saying 'grave' pull it off? Do they have the character? Do the have the context? The subtext? Do the rest of their words fit with 'grave', or will it stick out?
If you can answer those questions, you will know if you can use the word.