Although there is not a definitive answer from a Nintendo dev (which is probably the only way to get definitive proof), there has been a plethora of other players asking this question, and the consensus is no, it is not possible to enter this room, or any of the other small "outhouses" attached to buildings that don't have interactable doors/openings.
Here are a few conversations from other players confirming this theory...
- From BOTW Reddit:
If you pay attention you will see that there is a few rooms like that
one in the village and none opens. I guess its just cosmetic, at least
for now. Maybe they had to cut down rooms because the Wii U couldnt
handle so much stuff in the village.
I believe they're outhouses.
Of all the things Nintendo had to put in the game, I think they
decided to skimp a bit in certain areas.
From a GameFAQs thread, which also confirmed lack of entry.
From another GameFAQs thread, where multiple players explicitly indicate the lack of ability to enter "Link's shack":
There's also a "storage shed" kind of addition on the back of the
house - WITH A DOOR - that you can't open and get into ...