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Fix some algebraic curve $\gamma$ if natural number $\mathbb R^4$ such that for any 4 distinct points N$.Choose a half-circle $x,y,v,w$ we have \gamma$. Let $y-x+w-v\ne 0$.Consider all M$ be the number of integer points on $\gamma$One can choose $\gamma$ so that there are $M$ is quite many big;say $M\gg C\cdot N^{2/5}$ is easy to arrange, but one can do better.

Note that for any two distinct pairs $(x,y)$ and $(v,w)$ of these integer points .on $\gamma$ we have$$|(y-x)+(w-v)|\ge 1.$$

For each integer point $x=(x_1,x_2,x_3,x_4)$ on $\gamma$ x=(x_1,x_2)\in\gamma$, consider the number$$\bar x=x_1+(2\cdot N)\cdot x_2+(2\cdot N)^2\cdot x_3+(2\cdot N)^3\cdot x_4$$All $\bar x$'s form a a set $\bar X$ of integers in $[0,(2\cdot N)^4]$ N)^2]$ such that $$\bar y-\bar x+\bar w-\bar v\ne 0.$$

Now choose small for any two distinct pairs $\varepsilon>0$ (\bar x,\bar y)$ and mark $(\bar v,\bar w)$ of numbers in $\bar X$, $$|(\bar y - \bar x)+(\bar w - \bar v)| \ge 1.$$

Mark all points on a circle an with central angles of the form $\tfrac{\bar x\cdot\pi}{(2\cdot N)^2}$, $\bar x\cdot\varepsilon$x\in \bar X$.You all Note that the angles between the lines through these points will be $\ge \varepsilon$.

For tfrac{\pi}{2\cdot (2\cdot N)^2}$.

If you look for a set of $n$ points, you it should give a bound on sufficient to take $N\gg n^{5/2}$ and therefore the angle of the form will be $O(1/n^{2+?})$;an obvios upper bound is \gg 1/n^5$.This estimate can be improved, but I do not think you can get $O(1/n^2)$.\gg 1/n^2$ on this way;this might be the optimal asymptotic (?).
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I do not think you expect to find an optimal configuration. I will describe a general idea to produce a reasonable one. Hope someone will give you a better answer.

In the answer I looking for a configuration of points on the plane such that angles between lines through pairs of these points are big enough. This construction can be modified easily for your needs.

Take some algebraic curve $\gamma$ if $\mathbb R^4$ such that for any 4 distinct points $x,y,v,w$ we have $y-x+w-v\ne 0$. Consider all integer points on $\gamma$ with all coordinates in $[0,N]$. One can choose $\gamma$ so that there are quite many of these points.

For each point $x=(x_1,x_2,x_3,x_4)$ on $\gamma$ consider number $$\bar x=x_1+(2\cdot N)\cdot x_2+(2\cdot N)^2\cdot x_3+(2\cdot N)^3\cdot x_4$$ All $\bar x$'s form a a set of integers in $[0,(2\cdot N)^4]$ such that $$\bar y-\bar x+\bar w-\bar v\ne 0.$$

Now choose small $\varepsilon>0$ and mark points on a circle an angles of the form $\bar x\cdot\varepsilon$. You all the angles between the lines through these points will be $\ge \varepsilon$.

For $n$ points, it should give a bound on the angle of the form $O(1/n^{2+?})$; an obvios upper bound is $O(1/n^2)$.