The sampling and bit rates for audio clips in the 16-bit era had to be VERY small due to low memory footprints. So you can record "normal" audio and then export out the audio in a low spec setting, it should achieve the low-fi sound you're after.
Aside from that, if you wanted to keep the file specs higher, then you can work with EQ to lower fidelity and then a bit crusher distortion type of plugin to add in some of that crunchiness. I would advice doing an either/or approach here. Doing both at the same time would make it sound even worse than 16-bit era audio clips. These are just two quick ideas - there are probably other approaches that would work as well.