One can define the degree of separation between the classes of points based on the power means
\begin{equation}
M_p(X):=\Big(\frac1N\,\sum_1^N x_i^p\Big)^{1/p},
\end{equation}
where $X=(x_i)_1^N$ is a finite family of nonnegative real numbers and $p$ is a real number. As $p$ increases from $-\infty$ to $0$ to $\infty$, $M_p(X)$ increases from $M_{-\infty}(X):=\min_1^N x_i$ to $M_0(X):=(\prod_1^N x_i)^{1/N}$ to $M_\infty(X):=\max_1^N x_i$.
For sets $A$ and $B$ of points on the plane, a "distance" function $d$, and parameters $p,q,r,t,u,v$ in $[-\infty,\infty]$, define the $p$-size
\begin{equation}
s_p(A):=M_p((d(a,b)\colon a\in A,b\in A))
\end{equation}
of $A$ (one may take $p=0.5$ or even $p=0.1$ to decrease the influence of outliers);
the $q$-distance
\begin{equation}
d_q(a,B):=M_q((d(a,b)\colon b\in B))
\end{equation}
from a point $a$ to the set $B$ (one may take e.g. $q=-1$, to imitate the minimum, in a softer fashion);
the $(q,r)$-distance
\begin{equation}
h_{q,r}(A,B):=M_r((d_q(a,B)\colon a\in A))
\end{equation}
from the set $A$ to the set $B$ (one may take $r=0.5$); the $t$-symmetrization
\begin{equation}
H_{q,r,t}(A,B):=M_t((h_{q,r}(A,B),h_{q,r}(B,A)))
\end{equation}
of $h_{q,r}(A,B)$
(a $(q,r,t)$-generalization of the Hausdorff distance; one may take $t=1$); and the $(p,q,r,t,u)$-separation
\begin{equation}
S_{p,q,r,t,u}(A,B):=\frac{H_{q,r,t}(A,B)}{M_u((s_p(A),s_p(B)))}
\end{equation}
between the sets $A$ and $B$ (one may take $u=1$).
Finally, for classes $C_1,\dots,C_k$ of points, define the overall $(p,q,r,t,u,v)$-degree of separation
\begin{equation}
S_{p,q,r,t,u,v}(C_1,\dots,C_k):=M_v((S_{p,q,r,t,u}(C_i,C_j)\colon1\le i<j\le k))
\end{equation}
(one may take $v=-1$).
(For even more flexibility, one may use "distance" functions other than the Euclidean one. One can also use, instead of or in addition to the power means, the median and its various generalizations, as well as various trimmed and Winzorized means.)
As an example, for the four classes of points in each of the two pictures here, we have the values of the degree $S_{p,q,r,t,u,v}(C_1,\dots,C_4)$ of separation equal about $3.326$ and $1.872$, respectively, with $(p,q,r,t,u,v)=(0.5, -1, 0.5, 1, 1, -1)$ and the Euclidean distance for $d$. The second picture is obtained from the first (left) one by moving the blue set to the left and the yellow set down, both then closer to the green set; that decreases the overall degree of separation from $3.326$ to $1.872$.
