also: ふめんもく
He resigned from the company due to a disgraceful act.
試験に落ちて不面目に思う。
I feel ashamed for failing the exam.
面目ない is a more colloquial expression meaning 'ashamed' or 'sorry', whereas 不面目 is a stiffer, noun/adjectival noun form.
Standard kanji spelling; the reading ふめんぼく is more common than ふめんもく.
Less common reading; ふめんぼく is preferred in modern usage.
Compound of 不 (negative prefix) and 面目 (face, honor). The word 面目 itself comes from Chinese, and 不面目 follows the pattern of negating a noun to mean 'lack of' that quality.