"Back streets" here refers to the poorer quarters of larger cities, where these "missions" were/are set up to administer to the poor and/or homeless.

"Missions" in this sense were/are relatively small establishments-- the smallest usually in empty storefronts or commercial buildings, rather than in purpose-built structures.

Typically, they were set up to (at least) serve meals to the poor, but the larger ones could include dormitories providing a warm place to spend the night and, in recent times, various forms of counseling (for substance abuse, employment) might be available.

Religious services might or might not be included in the offerings, most, but not all, "missions" being funded by religious institutions, some being attached to churches in the neighborhood.

This 1955 film gives you the idea:

http://www.imdb.com/find?s=all&q=guys and dolls&x=0&y=0

Though there is quite a bit of parody in that comedy.

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Note added at 1 hr (2009-07-21 14:29:22 GMT)
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If an alternative term is needed (and I don't think that it is), how about:

"...the God of the whole earth, not just the God of the Temple, cathedrals or the most modest missions serving the poorest of the poor in the inner cities."

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Note added at 1 hr (2009-07-21 14:38:45 GMT)
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"most modest storefront missions" would work in U.S. English, since most of these places (at least in the past) have been in old "storefronts" rather than in purpose-built buildings.