僕は全然食べる気がしない。

。

I don't feel like eating at all.

Literal

I [topic-は] entirely eat feeling [subject-が] not-do.

気がする ('have a feeling, feel like'). ~気がしない ('don't feel like, no inclination to'). The dictionary-form verb 食べる modifies 気 attributively — 'a feeling of eating.' 全然 ('entirely') + negative = 'not at all.' A classic 'no appetite' expression.