One turn of the earth, and the span a life is lived in units of. Also, in this house, one continuous stretch that may run far past a date.
Old English dæg (day), from Proto-Germanic *dags; one turn of the earth, and by extension the unit a life is counted in.
Old English dæg named the turning itself, not the clock that later divided it, and this house's own stretched use of the word sits closer to that old sense than to any calendar square. A day here can outlast the date it started on, because the word was never really bound to the date at all.