 Polynomial vanishing on a finite set of spatial lines

A family of $L\ge 1$ lines in ${\mathbb{R}}^3$ imposes at most $L(d+1)$ linear conditions on degree-at-most-$d$ polynomials, by polynomial restriction to a line. The coefficient space has dimension $\left(\genfrac{}{}{0ex}{}{d+3}{3}\right)$. Whenever $(d+2)(d+3)>6L$, a nonzero common vanishing polynomial exists. Taking $d=\lceil \sqrt{6L}\rceil$ gives degree less than $4\sqrt{L}$.