An extremely simple object-size checker that can handle anything returned by JSON.

mem.py

from collections import Sequence, Mapping
from sys import getsizeof

def traverse(obj):
    if isinstance(obj, basestring):
        yield obj
    elif isinstance(obj, Sequence):
        yield obj
        for o in obj:
            for inner_o in traverse(o):
                yield inner_o
    elif isinstance(obj, Mapping):
        yield obj
        for k, v in obj.iteritems():
            for inner_k in traverse(k):
                yield inner_k
            for inner_v in traverse(v):
                yield inner_v
    else:
        yield obj

def get_total_size(x):
    unique = {id(o): o for o in traverse(x)}
    return sum(map(getsizeof, unique.values()))

if __name__ == '__main__':
    print get_total_size({'a': 'b', 'c': ['d', 'e', 'f'], 'g': {'h': 'i'}})

Warning:

This checks for duplicate objects when it calculates the memory usage, but it does not check for cycles during traversal and will explode if they are present.