B should be the answer.
Catch for such question is, read the whole sentence in one go (just forget that some part of what u read is underlined).
Part one: "In the traditional Japanese household, most clothing could be packed flatly."

Part B: "and so it was not necessary to have elaborate closet facilities."

You will feel a change in tone. So, clearly you will feel something is wrong in given sentence. Now, this point will be helpful only when you have practiced many SCs, else, you will not be able to get the point.
So, going by books



B: replaces "it was not necessary " with one word "unncecessary".
C "there was no" looks wordy, when you know that alternative short form is available in B.

D: "Being" is almost always wrong. "There being" is deadly combination. Avoid it.

E: There are 2 catches, oneis use of "as". I would prefer "so" here, as clause 1 describes some situation and clause 2 is implication of clause 1.

Second cartch for E is "no elaborate" gives me an impression that "though closet facilities are required, but peole dont get involved in using those facilities." Modifier is wrong.

Hence I go with B.

Option A.
It is important here to know what 'flat' and 'flatly' means
flat: without variations in height, flattened at the ends.
flatly: a manner that shows no emotions, in a manner showing completely certainty.

From the two definitions above we can see that the word most associated with the sentence is 'flat' thus, every option with 'flatly' is incorrect.

Option A is incorrect.

Option B.
This is the correct answer concise, well constructed and good parallelism.

Option C.
too wordy, it could have been more concise.This option is incorrect.

Option D.
awkward construction because of the use of 'being' the give no sense to the option for complementation.

Option E.
the first word here is wrong.Thus, this option is incorrect