If they are gas (you should know), then it could be a problem in your gas line (!!!). If a gas appliance runs with a pilot light, then they will also have a thermocouple... a little metal stick that's within the pilot flame; it can tell if the pilot is on based on if it's getting hot or not. It's a safety feature that's wired in so that if the pilot light every goes out, the gas is shut off. This prevents the gas from leaking into your home. If there is an issue with your gas line causing it to run out/lose pressure then the pilot lights would go out and the thermocouple would shut it off and then you'd be required to relight it. Based on how the fireplace has been working and still works when relit, yet the stove has had the issue, it sounds like the problem would be between there in the line. Although it could just be a coincidence that your stove thermocouple is more sensitive than the one in the fireplace. If you're able to tell that it's a consistent gas flow and maintains a decent size (watch the pilot) then it's probably your thermocouple that is going bad. You'd need to get a replacement.
If the gas is inconsistent, check to see if any gas valve is partially closed or contact the gas company and have them check the line and meter, it might not be an issue in your home, but rather from the outside.
As for the electrical side..
Since they're on separate circuits (I hope), then it's possibly just a coincidence that the fireplace is going out now, however it could be an underlying problem with the overall electrical system. Since the breakers aren't flipping, it's not shorting out (enough to catch them) and so the other option might be loose wiring or plug. Typically if there is shoddy electrical work in one area of a home, it'll follow suit that they did it everywhere. The other option would be, again just a coincidence, that your stove and fireplace have hit the end of their lifespan based on their age and usage. I'd recommend getting an electrician to come out and see the problem, they'd get a better understanding of what might be going on.. if it's electrical equipment and not gas.