The Lady of Shalott lives in a castle of “four gray walls, and four gray towers” on an island in a river. She is under a curse that obliges her not to pause her weaving to look upon Camelot through her window:
No time hath she to sport and play:
A charmed web she weaves alway.
A curse is on her, if she stay
Her weaving, either night or day,
To look down to Camelot.
“Stay” here needs to be understood in the archaic sense “leave off, cease, discontinue” (OED). The curse, however, permits her to look in her mirror:
Before her hangs a mirror clear,
Reflecting tower’d Camelot.
And as the mazy web she whirls,
She sees the surly village churls,
And the red cloaks of market girls
Pass onward from Shalott.
She incorporates these passers-by into her weaving, perhaps in the form of a tapestry:
But in her web she still delights
To weave the mirror’s magic sights
“Web” here means “a piece of fabric in the process of being woven” (OED), but also suggests a spider’s web, representing the net of magic in which she is entangled, or the dereliction of the tower in which she sits weaving.
So when she says that she is “half sick of shadows” there are a couple of ways to interpret this. Perhaps she means that she is sick of living in the shadows of her tower and wishes she were outside among the “market girls” and “damsels glad” of Camelot. Or perhaps she means that she is no longer satisfied with the reflections in the mirror and wants to see the true images for herself. This would be “shadow” in the sense, “the faint appearance of something seen through an obscuring medium” (OED). We have to remember that medieval mirrors were usually polished sheets of bronze or silver, in which the quality of the reflected image is much worse than in glass mirrors.
The version of the poem quoted in the question is the 1833 version; Tennyson later revised the poem; in the 1842 version it is clear that the “shadows” are the images in the mirror:
And moving thro’ a mirror clear
That hangs before her all the year,
Shadows of the world appear.
There she sees the highway near
Winding down to Camelot
The use of “half” in “half sick” indicates some ambivalence—to remain weaving in the tower is to live in a world of shadows, but the bright world outside is also dangerous. What is this danger? The poem is balanced between a literal interpretation, in which the danger comes from the magical curse, and a metaphorical interpretation:
The key to this wonderful tale of magic, and yet of deep human significance, is to be found perhaps in the lines—
Or when the moon was overhead,
Came two young lovers lately wed;
“I am half sick of shadows,” said
The Lady of Shalott.
The new-born love for something, for some one, in the wide world from which she has been so long secluded, takes her out of the region of shadows into that of realities. The curse is the anguish of unrequited love. The shock of her disappointment kills her.
Alfred Ainger (1891). Tennyson for the Young, page 113. London: Macmillan.
This interpretation explains why the Lady’s “half sick” remark comes on the heels of the “two young lovers lately wed”—she wishes that she too were newly wed. Hallam Tennyson says (Life and Works, page 151) that the interpretation was given to Ainger by his father.