Given a finite field $\mathbb F_q$, we say that a set $A\subseteq \mathbb F^n_q$ is an $(r,c)$ subspace evasive set if and only if for any affine subspace $U\subseteq \mathbb F^n_q$ with bounded dimension $\dim(U)\leq r$, the intersection size $|A\cap U|$ is at most $c$. The probabilistic method shows the existence of $(r,c)$ subspace evasive sets with $|A| = q^{(1-\epsilon) n}$ with $c = O(r/\epsilon)$. However, the best explicit constructions by Dvir--Lovett (STOC 2012) and Ben-Aroya--Shinkar (CJTCS 2013) have $c$ growing as an exponential function of $r$.

Here, we give an explicit construction of such subspace evasive set that achieves the intersection bound $c=O(\min(r\log(r/\epsilon)/\epsilon, r^2/\epsilon))$ when both $r$ and $1/\epsilon$ are constants. Our approach is simple and the key idea is to compose subspace design codes (Guruswami-Xing, STOC 2013) with a suitably chosen ``local'' subspace evasive set. As a corollary, we give the first construction of an explicit (strongly) capacity-achieving list-recoverable code with nearly optimal list size, circumventing all previously-known barriers to construction.