How do I find the Apple startup chime/sounds on my computer? I know there has to be someway for it to be stored, but I'd just like to be able to access the file/raw binary...
The sounds are stored in firmware, and not as a file.
Some more info regarding startup sounds here: https://support.apple.com/en-us/HT202768
Not stating your hardware makes this further a bit complicated. There were quite a few Macintoshes and Macs with very different sounds and different ROMs, firmwares…
If you just want to access the sounds (not extract them form your machine by hand, just use them): use MacTracker for all the old StartUp Sounds. In the package file they are stored in ´Mactracker.app/Contents/Resources/Chimes´ as m4a files.
For comparison and background info read this article and listen to the linked mp3 there.