*If* google fails, it won't be around one day and gone the next. In any case, there are other search engines that do a nearly as good a job as google, so we would still be able to do search. Migrating from Gmail would be slightly more complicated, but would still be doable, especially considering that all other free email providers would love to, and actively encourage and help anyone wanting to migrate. Generally, if google stumbles, there are plenty of others ready and willing to pick up the slack.
Far more worrying would to have a systemic failure of the entire 'free stuff & ads 'business model. If providing free search and/or email (and social networking) is no longer profitable, we are truly screwed. It is not that we would have to pay much for search and email (marginal costs are very small); but the net would probably balkanize (if search isn't free, why would content? And in this case, why link to your competitor's content?) and stovepipe. Using the Internet use would end up looking like using a mobile phone. Useful, no doubt. But a shadow of what its former self.
Sorry for the melodrama; but the current human architecture of the internet is very fortuitous, but was hardly inevitable. It emerged, largely unplanned, from a series of developments that could easily have happened elsehow. There are other ways of creating a world wide network, that would do almost all that the net does, but provide much more top-down control.