To besiege means to attack with an army, or to pester with many requests. When all your teachers ask you to hand in assignments on the same day, you can end up feeling besieged.
The source of the word besiege in its military sense is the Latin word for "seat." When an army settles down in front of a fort or other site of attack, they are besieging it or taking a seat there. Picture them continually bombarding the fortress with arrows and cannon shot, and you've got the picture of the figurative sense of the word, "to pester with requests, etc." If you get a lot of spam in your e-mail, you are besieged with advertisements — and maybe also besieged with worries that you'll get a virus!