Tiré par les cheveux

This somewhat painful expression literally translates to pulled by the hair.  You'll often hear it as an exclamation: c'est tiré par les cheveux!  The context, however, seldom involves a ponytail you could gently pull on.

Tiré par les cheveux means far-fetched or strained, in order to fit an agenda.  It is used to describe a situation, argument, or explanation that is hard to believe or that requires a significant stretch of the imagination. It implies that the situation is so improbable, unnatural or illogical, that it seems as if it has been forcibly pulled out of thin (h)air.

C'est tiré par les cheveux is fairly close to the expression ça ne tient pas debout - litterally it does not stand upright - which means it does not make any sense.  If your story is pulled by the hair, however, it makes some sense, in a bizarre and convoluted way.

Note that the past participle tiré agrees in gender and number with the subject it describes:
ton histoire est tirée par les cheveux - your story is far-fetched
leurs explications sont tirées par les cheveux - their explanations are far-fetched.

Origin

The expression has been in use since at least the 16th century but beyond having written records from that time period, it’s not very clear where the saying comes from or why exactly it’s connected to les cheveux - hair.  Some suggest it may have to do with how pulling someone by their hair forces them to do something they wouldn’t otherwise do.

There are reports of such torture back in the good old days which consisted of using such unwanted capillary treatments until people confessed whatever you wanted them to confess. Much like any other type of torture.  The victim would indeed say anything you wanted to hear, or anything they thought you wanted to hear, or anything whatsoever, as long as the torture stopped.

This explanation of the phrase's origin is in itself a bit tirée par les cheveux; not quite impossible, yet not totally convincing.  Such is life, sometimes.

Examples

Son alibi est tiré par les cheveux.  Pourquoi serait-il sorti au milieu de la nuit pour se promener en rase campagne sous la pluie? 
His alibi is not credible.  Why would he go out in the middle of the night for a walk in the open countryside in the rain?

« Il y a des figures claires et démonstratives, mais il y en a d'autres qui semblent un peu tirées par les cheveux, et qui ne prouvent qu'à ceux qui sont persuadés d'ailleurs. »
Blaise Pascal - Pensées
There are clear and demonstrative figures, but there are others which seem a little far-fetched, and which prove only to those who are persuaded. » 
Blaise Pascal - Thoughts