Partial access MongoDB databases without replication. Is that possible?
You haven't mentioned what driver you will be using, so I can't give you a direct link to the specific option in your language of choice, but yes there is an option to return partial results from a sharded cluster if there are shards down (I still don't advocate running without replica sets, but that's a different discussion).

You can set it in the shell as follows (sample values, will work for findOne also):

db.collection.find({}, {}, 0, 0, 0, DBQuery.Option.partial)


Each of the drivers should have support for setting this option, either directly or similar to the shell via an option.  See the partial option toward the bottom of the node.js cursor documentation for example or the DBCursor setOptions and associated QUERYOPTION_PARTIAL in Java.  For reference, the relevant feature requests to add partial results options are SERVER-2020 and SERVER-4610