I\'m trying to use the Google API v3 to access one google calendar and according to the documentation here : http://code.google.com/apis/calendar/v3/using.html#intro and her
You cannot access your calendar information using API Key. API keys (or simple API acess key) are not authorized tokens and can only be used for some API calls such as a Google search query etc; API keys will not let you access any user specific data, which I am assuming is your objective through this calendar application.
Also, from what I see in your code, you are creating a client object which is going to use OAuth 2.0 authentication and hence you are getting authentication error messages.