Sure, this is possible. 8 kHz sampling is only 125us between samples which is a fairly long time for the Arduino. 10 kHz sampling rates are achievable using the built-in A/D converter. You will need a microphone and amplifier, either on a microphone breakout board like this:
or one of our soon-to-be-released new products (which I don't want to pre-announce 
)
You are limited by RAM so you will definitely have to stream the data out as you get it. The bit rate is quite modest (10 bits at 8000 sps is only 80 kbit/s so could be done over a 115200bps serial link). You could even knock that down to 8 bits per sample (64kbit/s) and get quite intelligible speech.
--
The Arduino Drum Machine: 14-track MIDI drum machine sequencer / groove-box