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I feel like I've been bitten by my own pet dog.
Literal
Pet-dog [by-に] hand [object-を] was-bitten feeling is.
A casual use of a set expression. 飼い犬に手を噛まれる is a ことわざ (proverb) literally 'to have your hand bitten by your own pet dog,' meaning 'to be betrayed by someone you trusted,' 'to be stung by someone you took care of.' The image is of a dog you've raised and fed turning on you. Here the speaker says ~気分だ ('I feel like X'), framing the proverb as a simile for the current emotional state rather than literal betrayal. The passive 噛まれる marks the speaker as the affected party — classic adversative passive. ~気分だ ('the feeling is X') is a set predicate for subjective mood.