Is it viable to write a story taking place in four spatial dimensions?

How do you plan to define the dimensions, though? A common definition in fiction is to keep the first three as x,y, and z like normal, and the fourth one denotes the universe or plane you're in, which basically describes the multiverse. In that case, those four coordinates are actually spatial. But otherwise, we are living technically in a 4-D world where space and time cannot really be disentangled from one another.
 
Legit love this idea and discussion. It makes me think of Carl Sagan and his bit on tesseracts and apples:

 
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