To become so intoxicated after consuming alcohol and/or drugs that your body can no longer handle the toxicity and needs to get rid of it. The "(not so) elegantly" refers to the methods your body uses to expel the intoxicants (vomiting, etc)
My bestie got so trashed drunk last night at the party that she got nauseous and puked all over herself. Guess you could say that she was (not so) elegantly wasted.