The expression literally translates as being in nice sheets and, as such, could conjure up to the uninformed mind a picture of wellbeing and comfort. As language would have it, however, it means the exact opposite: être dans de beaux draps means to be in trouble, in a pickle, up the creek without a paddle or any tricky situation.
How can that be?
First off, the word draps means sheets in modern French (as in bed sheets), but until the 18th century, it referred to the fabric that clothes were made of and was used as a synonym for clothes.
Originally, the full expression was être dans de beaux draps blancs, i.e. to be in nice white clothes, and described a shameful situation. Now, why would the color white, associated with purity and innocence in Western culture, have had a connotation of shame in the past?
The connotation might stem from theatrical performances in the Middle Ages. It was common practice then to have a character dressed in white appear on stage as a target for the audience to shout insults at - the sole purpose was to include the audience in the show. The Church too adopted the symbol and made it customary for people accused of lust to attend mass dressed in white, so the white color would bring out the "black" aspects of their souls. By extension, until the end of the 17th century, putting a man in beautiful white clothes meant criticizing him, and being in beautiful white clothes meant you were subject to mockery.
Today the word white has disappeared, and the meaning shifted from shameful to vulnerable, or simply uncomfortable.
Je suis dans de beaux draps! Je viens de perdre mon portefeuille!
What a mess! I just lost my wallet!
Il n'a pas pris le temps de préparer sa présentation, maintenant il est dans de beaux draps!
He didn't take the time to prepare his presentation, now he's in a pickle!