coordinates - Is nearest search done on Euclidean plane accurate

I am writing a program which will find all the points within 1km. I am planning to to do this on a Euclidean plane as my database only supports Euclidean plane or sphere indexing (http://docs.mongodb.org/manual/applications/geospatial-indexes/). I chose the plane due to performance. I was wondering how accurate using this plane would be all over the earth. I am not too considered about EXACT matches, but I would like no more than 50m of error when searching for all points within 1km of a certain point. Would this work all over the earth or would I lose accuracy if I move throughout the earth (poles and stuff)?

PS: I am not expert at GIS stuff, wouldn't even consider myself a beginner.