Actually, some MP3 players can record audio. It depends on the player.
If your MP3 player cannot record audio, the next most likely candidate is your smartphone. You'll probably need to download an audio recording app because for some reason they don't seem to ship with one built-in. And you'll need a headphone/mic splitter which fits your phone's headphone jack. Connect an audio cable from the tuner's phono out jack to the splitter's mic. Then use the app to record.
A PC will work two, but (unless it's a laptop) would require an audio cable long enough to reach between the PC and tuner (which can pick up stray electrical noise if it's really long). Or lugging the PC next to the tuner.