There are different methods to sample from games.
The first and easiest is to hook up your console to your computer, to turn off the music and to do the in-game action that triggers the sound effects you want whilst recording the soundcard output with an audio recorder. This isn't foolproof though, for instance, Goldeneye 64 doesn't have the option to turn the music completely off.......
Another way is to download an emulator and ROM of the platform/game in question. Some games/roms allow you to access the game data which includes the sound effects. You can use an audio recorder to sample whichever ever samples tickle your fancy.... Using this method I was able to rip the street fighter 2 sound effects directly from the game long before Capcom did their free sample give away recently....!

Lastly, there are many hardcore game hacking communities out there (as mentioned above). Sometimes members of these communities find a way to rip/dump the entire game audio directly from the cart/cd into easily usable audio formats. This is often the case for games where audio files are not used, but where games use proprietary onboard sound devices & midi files to generate audio... PSX is one example of this, as well as N64 (I think....

)
One thing is for certain, if you're desperate enough for a certain sound from a game, searching the web long enough will always turn up results

I spent two years searching for a particular (secret) sound from a certain classic N64 title, and in the end I found a game hacker who shared with me the audio he'd ripped from the cart!

------------------------------
EDIT: You could also synthesise your own game sounds using many of the freeware synths designed specifically for the purpose. For recomendations on good 'game' synths, check out
nes-nintendo-old-school-game-consoles-and-game-noises-t71420.html