Another kind of answer involves the EHP sequence. If $\pi_n^k=\pi_n(S^k)/\text{odd torsion}$, then there are exact sequences $$ \pi_{n+2}^{2k+1}\xrightarrow{P}\pi_n^k \xrightarrow{E} \pi_{n+1}^{k+1} \xrightarrow{H} \pi_{n+1}^{2k+1} \xrightarrow{P} \pi_{n-1}^k. $$ (These are homomorphisms of abelian groups, except that when $n=k=0$ the map $H$ is the self-map of $\pi_1^1=\mathbb{Z}$ given by $m\mapsto m(m-1)/2$.) We also have the "boundary conditions" that $\pi_n^k=0$ for $n<k$, or for $n>1$ when $k=1$, and the fact that $HPE^2$ is multiplication by $2$ on the group $\pi_{4n-1}^{4n-1}=\mathbb{Z}$. This provides a rather intricate pattern of connections between all the groups. You can try to write down a system of groups and homomorphisms satisfying these conditions, without worrying about whether they are actually the same as the real homotopy groups of spheres, and you will find that it quickly gets very complicated.
There are two partial solutions that can be written down explicitly. The groups $\mathbb{Q}\otimes\pi^n_k$ are known: there is a copy of $\mathbb{Q}$ for $(n,k)=(i,i)$ or $(4i-1,2i)$, and everything else is zero. It is easy to write down the maps $E$, $H$ and $P$ in this context and to check that everything is exact. There is a version of the $\Lambda$ algebra that is a relatively simple structure with properties similar to those specified above, but it has $P=0$ and each group $\Lambda_n^k$ is an infinitely generated $\mathbb{Z}/2$-module. (The $\Lambda$-algebra also has a differential, and it can be regarded as the $E_1$ page of an unstable Adams spectral sequence converging to $\pi_*^*$.)
I do not know any direct construction of a system of finitely generated abelian groups and homomorphisms with the required properties, and I do not think that anyone else does either. My guess is that any such system is of similar complexity to the homotopy groups of spheres.
Here is another interesting fact that sheds some light on this circle of ideas. Suppose we have systems $A_*^*$ and $B_*^*$ as above, and a morphism $f\colon A\to B$ that is compatible with $E$, $H$ and $P$, such that $f\colon A_1^1\to B_1^1$ is an isomorphism. One can then show that $f$ is an isomorphism in all bidegrees. This is another illustration of how tightly everything is linked together by the EHP sequence.