Many people use Google's DNS servers, so tons of cumputer illiterates would be unable to connect to any website. (On the top you might need connection to find another DNS server.)

Google provides caching DNS servers around the world for the same IP addresses. (quite famous ones: 8.8.8.8 and 8.8.4.4) At the end of November, their servers went down for 2 hours in my country. NIX.cz - the associacion of ISPs - reported that the traffic dropped drastically to about 20% of normal. It also affected some Andriod phones with Chrome browser.

Additionally Google provides many services outside their websites. For example many websites use reCAPTCHA and because it load code from Google, these websites would be broken, it includes many payment servies and online bankings.

Another notable case is that JavaScript-heavy websites sometimes use Google to load you the JS framework code - it's from the same source across multiple websites so you don't need to download it again.

So it would be a massive breakdown. Google is so much more than just search engine and mail.