Drew and Jeanne Barber thought they had found a heavenly place to raise their two young children when they toured a four-bedroom, 4,900-square-foot house on 11 acres in Farmington, Conn., in 2018.

A 1975 house with a gambrel roof that had been expanded with multiple additions, it offered plenty of space to spread out, manicured gardens, a cute stone pool house for the summer and — most important for Mr. Barber, an avid hockey fan — a big yard capable of accommodating an ice rink in the winter.

But it also had something less desirable that the couple didn’t discover until it was too late: a thriving colony of mice.