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Street was not used in the modern sense when surnames were being adopted. A street in a medieval English town was called a gate, cognate with the German gasse. Think of Ludgate, London, or Coppergate, York. The word street identified a Roman road, and has the same origin as the word straight (I think). There are some some English villages called "Street", all beside Roman roads, according to Paul Reaney. So someone given the name Street was likely to have originated in a settlement in the vicinity of one of those ancient roads.
However, there is another possibility, in my opinion. Street may identify an abandoned infant found in the street. This would have occurred after the word came to be used in its present sense. I've seen no record of the name being bestowed in this way, but I know that foundlings were often called after the place where they were found. So a child might be saddled with a name like, for example, John Threadneedle Street. I'm sure that John would have been eager to shorten his surname to Street at the earliest opportunity.