blind alley

from The American Heritage® Dictionary of the English Language, 5th Edition.

• noun An alley or passage that is closed at one end.
• noun A mistaken, unproductive undertaking.

from Wiktionary, Creative Commons Attribution/Share-Alike License.

• noun Used other than as an idiom: see blind,‎ alley. A street or passageway that leads nowhere.
• noun figuratively A course of inquiry that leads nowhere.

from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.

• noun a street with only one way in or out
• noun (figurative) a course of action that is unproductive and offers no hope of improvement