By "story tapes", I assume you mean audio books. The short answer is yes you can.
The longer answer is:
if they are in Audible format, and you transfer them using the Audible manager, you can simply open them in the same way as you would a text book, and you get an on-screen interface with "Play", "Pause", etc. and if you close the audio book, the Kindle will remember where you'd got to.
If they are mp3 files, they will play using the "experimental" mp3 player, over which you have no control except to start and stop playing.
They will, of course play in the voices that were recorded - the only time you'll hear a computer generated voice, is if you get your Kindle to "read you a story" from a normal text-based book.
To be frank, although I love my Kindle for reading, I don't recommend it for playing audio books. For that, I use a Sony mp3 player.