Record numbers of homeless families are overwhelming the state’s emergency shelter system, filling motel rooms at the cost to taxpayers of tens of millions of dollars a year.

An average of 2,100 families a night — an all-time high — were temporarily housed in motel rooms in October, just about equaling the number of families staying in emergency shelters across the state, according to state Department of Housing and Economic Development.

The demand for shelter is so great that the state has been temporarily shipping homeless families from Boston to motels in Western Massachusetts, although state officials said many have been relocated back again, closer to home.