I think you would probably want a car. It is the 'burbs after all. But you may be able to get by for a while bumming rides and taking public transportation.
Car or not - it's one of the big questions.
I'd say get/bring a car if you can. Things are spread out, which means decent parking around Stanford but hard to get places without a car. Also, public transportation is nothing like east coast metro areas.
As an undergrad at Stanford, freshmen weren't allowed to have cars, so I'd bum rides off of people with cars. It's totally do-able, so if you don't have a car don't worry about it. Plus, it's a good excuse to make friends ;-)