Your city is under siege if it is surrounded on all sides by an opposing force on attack. Think of a castle surrounded by a legion of armed knights.
Like many military words, siege can be used metaphorically. If you start getting thousands of e-mail messages trying to sell you canned meat, you might feel like you're under a siege of spam selling spam. In this case, you have been besieged by spam. And even more unfortunately, if you are having a siege of bad luck, you have been besieged by bad luck.