I am trying to find a vector field in two dimensions which points in the negative radial direction and has magnitude 1.
Obviously the gravitational field on Earth's surface looks like that but how can we find such a vector field? To have a magnitude of 1, I guess that $x^2 + y^2 = 1$ where $x$ and $y$ are the vector components, but how do we get to the vector field definition?
Use polar coordinates. Let $\hat{r}$ be the radial unit vector, then:
\begin{align*} \vec{F}(r) = -\hat{r} \end{align*}
is the required field. Now $\hat{r} = \cos \theta \hat{i} + \sin \theta \hat{j}$. Also, from basic trigonometry, you know: $$\cos \theta = \frac{x}{\sqrt{x^2 + y^2}}$$ and $$\sin \theta = \frac{y}{\sqrt{x^2 + y^2}}$$
So in Cartesian coordinate the field you want is: \begin{align*} \vec{F}(x,y) = -\left(\frac{x}{\sqrt{x^2 + y^2}} \hat{i} + \frac{y}{\sqrt{x^2 + y^2}} \hat{j}\right) \end{align*}