Can you get a shorthaired golden retriever?

No. Goldens have medium-length hair with varying amounts of feathering. (Feathering is longer hair on certain spots ... google some images of Golden Retrievers and you will see differences in the amount of feathering.)

If you want a short-haired retriever get a Labrador Retriever .. they have short hair all over without feathering. (Labs are more energetic and somewhat less easier to train than Goldens though.)

Just a thought .... but if you are looking for a "shorthaired Golden" because you think short-haired dogs shed less .... that is NOT at all true!! That is a persistent myth for some strange reason but it has absolutely NO basis in reality. Short hair does NOT mean the dog will automatically shed less.

In fact, most short-haired Dog Breeds shed far MORE than Goldens ... examples are Dalmations and Corgis. Both those breeds shed FAR more than Goldens ... they just shed a lot of shorter hairs! Another fact is that most non-shedding breeds also happen to have the LONGEST hair ... like Maltese and Shih Tzu breeds.