day when prostitutes were obliged to take clients
Archaic term from the Edo period. On such days, prostitutes had to receive clients or pay a fee to avoid doing so. This sense is historical and not used in modern Japanese.
Today is a holiday, so let's go out as a family.
In the pleasure quarters of the Edo period, on 'monobi' days, prostitutes had to take clients.
Compound of 物 (mono, 'thing') + 日 (bi, 'day'). The exact historical derivation is uncertain; the spelling is conventionally associated with the meaning 'special day'.