Such a thing would not be a field.

You can define an additive group $\frac{1}{n}\mathbb{Z}$ if you like. However (for $n > 1$) it would not even be a ring, because it would not be closed under multiplication. (It's closure under multiplication would be $\mathbb{Z}[\frac{1}{n}]$, which would not have a smallest positive element, contrary to your design criterion.)

(Of course, you could define a partial multiplication on it. I don't think there's a good name for such a thing. I guess you could just call it "a subgroup of the rational numbers under addition, equipped with a partial multiplication operation that is defined and agrees with the usual multiplication on rational numbers when the result would still be in the subgroup")