The number of states required for a full rank state space model is the same as the number of distinct energy storage elements (i.e. distinct inductors and capacitors). Typically, the states are the currents through the inductors, and the voltages across the capacitors.
Let \$\small x_1=\$ current down through the 2 H inductor; \$\small x_2=\$ current down through the 1 H inductor; and \$\small x_3=\$ voltage across the capacitor (+ at top).
Nodal analysis gives,
\$\hspace{60mm}\small 8\dot x_1 -\dot x_2=3v_i-3x_1\$
\$\hspace{60mm}\small 2\dot x_1 -4\dot x_2=3x_2-3x_3\$
And, for the third differential equation, KVL on the right-hand LCR mesh gives,
\$\hspace{60mm}\small \dot x_3=2\dot x_2-2x_3\$
Solving for the state derivatives,
\$\hspace{60mm}\small \dot x_1=\left(4v_i-4x_1-x_2+x_3 \right)/10\$
\$\hspace{60mm}\small \dot x_2=\left(2v_i-2x_1-8x_2+8x_3\right)/10\$
\$\hspace{60mm}\small \dot x_3=\left(4v_i-4x_1-16x_2-4x_3\right)/10\$
which can be used to construct the state space equation,
\$\hspace{60mm}\small\pmb{ \dot x}=\pmb{Ax}+\pmb{B}u\$
For interest, the \$\small\pmb{A}\$ matrix is non-singular, and has eigenvalues,
\$\hspace{60mm}\small \lambda_{1}\approx -0.36\$
\$\hspace{60mm}\small \lambda_{2,3}\approx -0.62\pm j1.13\$