For instance, I run qemu all the time to emulate various android devices.
I suspect that you're thinking specifically of game emulation though.
Running software on an emulated device is fine legally as long as it doesn't violate copyright law.
For instance, you can legally backup software that you own in the US [1] - that extends to games as well - and because emulators themselves are legal (although you may also need to backup the device's BIOS), you can have a completely legitimate archive of copyrighted games to run via an emulator.
That said, it's unlikely that most people archive software themselves, and it is not legal to distribute backups in the US, even if both parties have legitimately acquired copies of the source material.
