This answer describes all of the solutions to the desired equation. In particular, it proposes three fundamentally different classes of solutions, with representatives $(a,b,c,d)$ given as $(0,−8,9,−1)$ and $(1,7,-7,-1)$ and $(-728,728,1,999)$, each among an larger class of similarly constructed examples.

Let $a+b=x^3$ and $c+d=y^3$ and $a+b+c+d=z^3$. Then
$$x^3+y^3=z^3.$$
This is impossible for $z\neq 0$ by Fermat's Last Theorem (or special cases thereof which Fermat actually did prove). So, we need one of the summands to be zero.

**Case 1:** $a+b+c+d=0$

In this case, we get that both $a$ and $d$ and $a+d$ are perfect cubes since their negatives $b+c+d$ and $a+b+c$ and $b+c$. So, now we need that one of $a$ and $d$ and $a+d$ is zero.

**Case 1a:** $a=0$

Now, we get that $b$ is a perfect cube, since $a+b$ is. However, then one of $b$ and $c+d$ and $b+c+d$ must be zero. However, it must be that $b+c+d$ is zero, since $b=-c-d$ cannot be zero since $a$ is. At this point, we can solve the system
$$a=0$$
$$b+c=\alpha^3$$
$$c+d=\beta^3$$
$$b+c+d=0$$
which has solutions for all $\alpha,\beta$. For instance $(0,-8,9,-1)$ is of this form.

**Case 1b:** $d=0$

This case, by symmetry, will yield the same result as the previous case, except with $(a,b,c,d)$ swapped to $(d,c,b,a)$.

**Case 1c:** $a+d=0$

Note that this implies $b+c=0$ as well, so $a=-d$ and $b=-c$ follow. Moreover, since $a+b+c=a$ is a cube, $a$ must be a non-zero cube. Then, we get the following family of solutions
$$a=\alpha^3$$
$$a+b=\beta^3$$
$$c=-b$$
$$d=-a$$
which can be solved for all $\alpha,\beta$ input into the system. For instance, the tuple $(1,7,-7,-1)$ is of this form

**Case 2:** $a+b=0$

Since $a+b+c$ is a perfect cube, we get that $c$ is a perfect cube. It suffices to have $c$ and $b+c$ and $c+d$ and $b+c+d$ to all be cubes. In particular, if you have a solution to $$\alpha^3+\beta^3=\kappa^3+\gamma^3$$
then you can generate a solution as
$$c=\alpha^3$$
$$b+c+d=\beta^3$$
$$b+c=\kappa^3$$
$$c+d=\gamma^3$$
$$b=-a$$
Which looks overdetermined, but one can see that the first four equations are linearly dependent, so is not overdetermined. More work is necessary to finish classifying this class of solutions. One can use an identity that Ramanujan was fond of $1729=1^3+12^3=9^3+10^3$ to get the solution $(-728,728,1,999)$. It should be noted that $\alpha,\beta,\kappa,\gamma$ must be distinct for this to give distinct $a,b,c,d$. I'm not sure how to characterize the general set of such solutions, but it is notable that one can plug in [taxicab numbers](https://en.wikipedia.org/wiki/Taxicab_number) to get a lot of solutions of this form.

**Case 3:** $c+d=0$

This is, by symmetry, the same as Case 2.