some digital mixers might have a hard drive source in them...but most analog and digital mixers you need to connect to a storage medium of some sort (ie. tape, digital tape, computer with hard drive, etc).
You might be confusing a mixer with a standalone recorder. Standalone recorders usually have mixing features (wouldn't be very useful without them) and can have a similar appearance to a mixer (this used to confuse me when I *first* started looking at recording gear. )
But a mixer, as benny said, is used to mix (pan stuff from one speaker to the other, adjust volume on each track, etc) and has outputs that you connect to the media; hard drive, tape, etc where the mixed signal is recorded.
As an aside - the question could be "Do I need a hardware mixer at all, or can I do it all on a PC?"
You may find that the answer is that you don't need a hardware mixer at all. If you are recording one input at a time, and then later mixing things at leisure, you could do it all on the PC and probably save yourself some signal-chain grief.
Don't get me wrong - the only reason why I wrote the above is because I have often seen your question asked by people who have been told that they *have* to have a hardware mixer to record stuff.