“être flingué” generally means to be exhausted. Note that this is an extremely familiar expression and can’t be used in a professional context (unless you are at ease with your colleagues).

More specifically, “être flingué” can also mean that you are drunk or under the effect of a drug depending on the context.

An example of a use of “être flingué” would be :

“J’ai super mal dormi, je suis flingué“

roughly meaning

“I slept really poorly, I’m exhausted“

In argot (the French language used by the underworld), a flingue is a gun. Hence, the verb flinguer means “to shoot at someone with a gun”, or “to kill someone with a gun”.
Ex:

Le terroriste a tenté de fuir, et la police l’a flingué.

Used in familiar usage, it can be used figuratively, and it then means either:

• to break something
  

  J’ai tellement joué à la Playstation que j’ai flingué la manette

• to criticize heavily someone or something or to give a bad rating
  

  Les professeurs ont tous flingué son travail de fin d’étude.
  Le directeur des ventes a flingué les vendeurs qui n’avaient pas atteint leurs objectifs.

Être flingué is the passive form of the verb and can be used for any of the meanings above.