General:
* Surround
* Encircle
* Hem in
* Close in on
* Overwhelm
* Inundate
Military:
* Lay siege to
* Blockade
* Assault
Figurative:
* Overwhelm
* Swamp
* Flood
* Assail
Other:
* Harass
* Pester
* Torment
* Mob
* Crowd
The best synonym for "besiege" will depend on the specific situation and the nuance you want to convey.