No, this isn't a legal requirement but it's strongly advisable for any business, even those just employing one person. A staff handbook makes clear the standards, rules and expectations which support the contract of employment and is instrumental to operating effective HR arrangements and promoting a harmonic and consistent workplace.
It's very hard (and quite possibly unfair) to expect employees to abide by particular rules and policies if they don't know what they are and this could also cause issues for an employer who might need to discipline an employee for allegedly breaking the rules they were not aware of.
A detailed staff handbook of essential policies and procedures is a must for any business.