SHA-256 is used by all the world, banks, governments, companies etcetc. If it get broke...well we can easily switch to something else with a client update. Meanwhile the entire world would collapse 
Most systems can switch out one hashing system for another. For example, when md5 was shown vulnerable to collisions SSL signatures simply switched to another hashing method.
Bitcoin has the disadvantage of being set in its ways. The majority of clients would have to be updated to at the same time switch to another method. Even if such a thing could be coordinated and the bitcoin contract ammended in the wild it would take a lot of time to organize it. Those that failed to update would reject these new blocks.