In The Miracle Worker, Annie explains that the experience in the garden house will be a siege. What does she mean by this both literally and symbolically?

Annie proposes to Helen's parents that she and Helen should live together alone in the garden house. This way, it is hoped, Helen will come to depend on Annie as her sole authority figure. She won't be able to do what she does now, which is to run off to her parents when Annie tries to teach her how to spell new words.
The new arrangement is a siege for Helen in the literal sense in that she'll have nowhere else to go. In history, when a city was under siege from invaders, its inhabitants would be trapped inside the city walls, unable to go anywhere else for fear of either being killed or letting the besieging forces inside the gates. Things may not be that desperate for Helen, but she'll certainly be besieged inside the garden house if Annie gets her way.
On a figurative level, Helen is under siege in the garden house in that she won't be able to obtain help from anyone else but will instead depend entirely upon Annie.