Answer: In this specific case, the answer is no, the family is not eligible, but only because they are not staying at the hotel “due to lack of adequate alternative accommodations,” per the McKinney-Vento Act. 42 USC §11434a. The only caveat is if the conditions of the motel room are such that they are inadequate – i.e. over-crowding, infestations, mold – the same criteria you would use to make any determination of inadequate housing under McKinney-Vento.
In other cases of families in motels, if they are there due to the lack of adequate alternative accommodations, then they remain eligible, even if the stay is long-term.