When the speaker says, "I am all longing," she means something like: "The very existence of my being is yearning for that which I do not have." In other words, she is deeply unhappy; she does not have anything that she wants and seems to acknowledge that she never will.
This notion is reinforced in the setting of the poem which is "a woody grove, under an oak-tree" in a "earthen cave." The speaker tells us that her life underground reminds her of "all [her] friends" who "dwell in the dirt." In other words, all of her friends are dead, and being underground just reminds her of that disturbing fact.
Her daily life in the poem appears to revolve only around mourning. She tells us:
There I may sit a summer-long day,
where I can weep for my exiled path,
my many miseries—therefore I can never
rest from these my mind’s sorrowings
While some might argue that the speaker is being hyperbolic, her tone is quite seriousness. This is a women afflicted by severe melancholy, and she cannot seem to escape it. All of this reinforces the embodied language of "I am all longing."
The line "I am entirely longing—" from "The Wife's Lament" is reinforced by the circumstances and setting in which she lives. Her husband, whom she loved and felt a great affinity for, has left her. He has ordered her to stay behind and live in a cave in a grove, and she feels betrayed because they had pledged their undying love to one another.
She is surrounded by the graves of former friends in a sort of wasteland. Every day she weeps because of her abandonment and isolation. She thinks of her absent husband and reflects on the idea of people who are separated from the ones they love—she seems to express that she hopes that he is suffering, too. She has concluded that he has plotted against her. She is "all longing" because she pines for what she does not have—and likely never has had.
The general storyline of "Wife's Lament" is that a woman has been married but, due to some trouble with his family, is now outcast. This is her lament, as the title suggests, as she is now living in "friendless exile." After being forced to leave by her husband's family, she is wandering and alone. The line to which you refer comes right in the middle of the actual setting in which she finds herself (though it's translated a bit differently in the version to which I'm referring).
They forced me to live in a forest grove,
under an oak tree in an earthen cave.
This earth-hall is old, and I ache with longing;
the dales are dark, the hills too high,
harsh hedgesoverhung with briars,
a home without joy. Here my lord's leaving
often fiercely seized me.
Her setting is stark and cold and lonely; it's not surprising that she finds herself full of nothing but longing. The next lines tell of her wandering through the forest from dawn to dark, pining for her lost love. The sterility of her environment matches the sterility (lack of love) of her heart. She is longing, and her surroundings have nothing to offer.
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