I currently don't have a dryer, so I have to hang my clothes on drying racks. I also have to do this inside, since I don't have much of a yard outside. How can I make a room less humid, especially on rainy or generally humid days, so the clothes dry faster?
Removing humidity from the air is an energy intensive process. Further, while your clothes will dry faster in a lower humidity environment, they will still dry in a humid environment.
Increasing the airflow, even on humid days, will significantly speed up drying. The clothes are still more damp than the air, so moving more air past them will remove water from them more quickly.
Fans are much less expensive to operate than dehumidifiers and heaters. Put the clothes in an area where you can move air into and back out of the room. If you have two windows in the drying room, that would be best - have one fan pulling air into the room via one window, and the other fan blowing air out of the room via the other window.
If you only have one window, pull air from the door into the room, and then blow it out the window.
If your room doesn't have two openings to increase the airflow, set up an oscillating fan near the one opening that alternately blows air out of the room and pulls air into the room.
Alternately, if you have forced air heating/cooling, you can turn the fan of that system on and it will circulate the air through the room into the rest of the house, where you can ventilate the more humid air outside.
An electric dehumidifier would reduce humidity, but would not necessarily be more efficient than a clothes dryer.
Calcium chloride is sold as a dehumidifier, but it quickly "fills up" with water, forming a semi-liquid mess, so it would be of no help for your application.
If you have a furnace-room where there is a hot water heater or furnace that runs most of the time, humidity is usually lower there because of the heat and the airflow.