A short term rental may be especially difficult to find near Stanford. Further away there are a number of businesses catering to business clients who stay for a few months. I don't remember the names for certain but I believe extendedstayamerica.com or suiteamerica.com could get you something.
Our rental market is very difficult, with very limited choices and low vacancy. There is no central source for finding the information, so you need to search the web, especially sites such as Craigslist, Zillow, MLSListings and also check property management companies. Good luck. It will require patience and persistence.