How crucial is location for the success of a startup?

Truly depends on the type of venture. Restaurants and retail need to focus on population density and traffic flows. Some professional providers (IT, insurance) can work from home and use 'rental offices' (Regus ets) to meet clients. Service providers (plumbing, a/c etc) - doesn't matter at all.

What type of business are you asking for?

It very much depends on the type of business for the start-up. But remember that a start-up usually has no brand recognition. As such, a store front on Main Street will always be better than an obscure location.

One other issue to consider is where the pool of talent lives. I am in the Bay Area and technical talent is in short supply and since much of this talent is young there is now a preference for being in walking distance of an urban core. As such we are seeing many companies moving from Silicon Valley to San Francisco in order to attract the talent.

Location, Location, Location - is eponymous to the success of business and as the others have stated so eloquently, location comes in many forms...

A business needs infrastructure, hence selecting a location depends on
Where founders willing to work under startup conditions are
Where the investors are
Where the markets are
Where skilled employees are
Where complimentary / support businesses are
Where communications match the business needs eg Freeways, airports, etc
Where customers are