A person is shut in with their own company and a world they can hardly stand to hear about. The speaker sings to the house, which is a quietly sad image: there is no audience but the walls, and "all alone" sits on its own line to stress it. Singing seems to be a way of filling the silence, or of keeping some warmth going when no one else is there.
The second stanza says it plainly: "I cannot bear / the news these days." There's no explanation of which news, which makes the feeling broader. It could be any bad headline, and that's the point. The last lines move outside, where the speaker walks in the garden "till my eyes grow over." The phrase is odd and a little ambiguous. It might mean tears, or a kind of glazing, a numbness that settles in. Either way, the mood is tired but gentle. The title, "Hanging in there," sounds cheerful, and the poem shows how thin that cheer is, and how small acts like singing and walking help someone keep going.