Give example of 2 standard uniform random variables with given Pearson correlation

I think there are a few approaches to generating random variables with given marginal distributions and specified correlations, but, in general, the ones I've seen all seem to be ad hoc.

One such method is to do the following: Let $U_1$ and $U_2$ be $U(0,1)$ i.i.d. r.v. and let $B$ be a r.v. distributed according to Bernoulli($0.25$). One way to construct $X$ and $Y$ as you've described is to set $X=U_1$ and $Y=BU_1+(1-B)U_2$. Intuitively, $Y$ 'copies' the same value as $X$ $25%$ of the time, and the other $75%$, they are independent uniform random variables. You can check that $X$ and $Y$ have the desired marginal distributions and correlation.