Assuming that you have $N$ electrons and $K$ spatial orbitals, the total number of electron configurations that can be built is $$ N_{\rm confs} = {K \choose N_\alpha} {K \choose N_\beta} \approx {K \choose N/2}^2 $$ since typically $N_\alpha \approx N_\beta$.
The exact solution is found by diagonalizing the configuration interaction (CI) Hamiltonian matrix, $H_{IJ} = \langle I | \hat{H} | J\rangle$. Although the matrix is typically huge (this is an understatement!), $N_{\rm confs} \times N_{\rm confs}$, it is also extremely sparse due to the Slater-Condon rules; the matrix elements between any two configurations that differ by more than a double excitation vanish. Moreover, the Hamiltonian is diagonally dominant, because the diagonal elements $H_{II}$ are single-configuration (i.e. Hartree-Fock) energies.
As a result, the lowest eigenstates of the full CI Hamiltonian matrix can be routinely determined with iterative methods; the famous Davidson method was designed for this purpose in the 1970s. Thanks to development of computers and algorithms, matrix sizes of a billion $\times$ a billion became feasible in the late 1980s, see Olsen et al.
However, due to the unforgiving exponential scaling of the number of configurations, the limits have not moved much since then. Tabulating the number of configurations obtained for $N$ electrons in $N$ orbitals, typically denoted as ($N$e,$N$o), one finds the following
\begin{array}{ll|ll}
N & N_\text{confs} & N & N_\text{confs} \\
\hline
\hline
1 & 1 & 11 & 2.1\times10^5 \\
2 & 4 & 12 & 8.5\times10^5 \\
3 & 9 & 13 & 2.9\times10^6 \\
4 & 36 & 14 & 1.2\times10^7 \\
5 & 100 & 15 & 4.1\times10^7 \\
6 & 400 & 16 & 1.7\times10^8 \\
7 & 1225 & 17 & 5.9\times10^8 \\
8 & 4900 & 18 & 2.4\times10^9 \\
9 & 15876 & 19 & 8.5\times10^9 \\
10 & 63504 & 20 & 3.4\times10^{10}
\end{array}
The (18e,18o) problem was thus feasible in the early 1990s. However, due to the unforgiving exponential scaling of the number of configurations, the limits have not moved that much since then. If you look at the literature, people often write (depending on the code and hardware at their use!) that full CI is limited in practice to (14e,14o) or (16e,16o) or (18e,18o).
The largest full CI calculations I know were reported in J. Chem. Phys. 147, 184111 (2017). The authors write in the abstract that (20e,20o) is routine with their new implementation and that they did a (22e,22o) calculation and a single CI iteration for the (24e,24o) problem; however, the text in Section V looks like only a single CI iteration was attempted for the (22e,22o) problem as well.
So, in summary, FCI is still limited in practice to somewhere around (16e,16o) - unless you want to waste a lot of computational resources to correlate a few more electrons.
The exponential scaling wall can, however, be pushed back a bit by being smart and leveraging the baked-in sparsity of the FCI problem with stochastic or selected CI methods developed by various authors. (You can see our recent paper here)
Another round is, of course, to reformulate the theory. We've done approximate (192e,192o) CASSCF for polyacenes with subtensor truncations of CCSDTQ56, see Mol. Phys. 116, 547 (2018). For the (50e,50o) pi-space calculation, we captured over 95% of the exact correlation energy. The full-valence calculation for 12acene just adds in sigma correlation, which is more localized and thus should be pretty much exactly captured by the used method.
