I doubt that busia's unit is frost free or self defrosting.
busia, has that thing been converted to modern refrigerant? It originally used Methyl Formate, which is explosive and can cause, among other things;
eye, nose, skin irritation; chest tightness, cough, labored breathing, shortness of breath, bronchitis; visual disturbance; headache, dizziness, unconsciousness; drying and cracking of the skin.
Aside from cleaning the coils, checking electrical connections and temperature control, if it has one, you need to get in a pro who can safely deal with and replace MF if needed.
Since it wants to run all the time, it could be temperature control thinks box is not cold enough or is stuck, so stick thermometer inside. In this situation faulty control would make box way too cold.
Dirty coils, worn compressor, or low gas would prevent unit from cooling, thermometer would read too low.
Gas is still available, equipment to handle it and person who has it may be difficult to find, replacement parts also rare. There are people who collect these things, antique shops may know of repair person.
One of these images came from Flickr, http://www.flickr.com/photos/27309380@N06/5012521936/
poster has lots of old reefers and ice boxes, probably a good source for info. I dunno if you can contact him without Flickr account, if not PM me and I could possibly act as intermediary.