I am uncertain how to perform some steps in this state-space realization.
Given $G(s) =$ \begin{bmatrix} \frac{1}{s^3}-\frac{1}{s+1} & \frac{1}{s^2}+\frac{1}{s+1} \\ \frac{1}{s^2} & \frac{1}{s} \end{bmatrix}
I can split the matrix into
$$ G(s) = \begin{bmatrix} \frac{1}{s} \\ 1 \end{bmatrix} \frac{1}{s} \begin{bmatrix} \frac{1}{s} & 1 \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \end{bmatrix} \frac{1}{s+1} \begin{bmatrix} -1 & 1 \end{bmatrix}. $$
Then I do not know how to achieve the next two steps
$$ G = \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} + \begin{bmatrix} -1 & -1 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} $$
And this step to the state-space realization
$$ \begin{bmatrix} A & B \\ C & D \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 & -1 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix} $$
The numerical matrices are in a compact notation form of the $A,B,C,D $ matrices.
I think you are making a mistake where you are combining two parallel systems:
$$ \begin{cases} \boldsymbol{\dot x}_1=\boldsymbol{A}_1 \boldsymbol{x}_1+\boldsymbol{B}_1 \boldsymbol{u}_1 \\ \boldsymbol{y}_1=\boldsymbol{C}_1 \boldsymbol{x}_1+\boldsymbol{D}_1 \boldsymbol{u}_1 \end{cases} $$
$$ \begin{cases} \boldsymbol{\dot x}_2=\boldsymbol{A}_2 \boldsymbol{x}_2+\boldsymbol{B}_2 \boldsymbol{u}_2 \\ \boldsymbol{y}_2=\boldsymbol{C}_2 \boldsymbol{x}_2+\boldsymbol{D}_2 \boldsymbol{u}_2 \end{cases} $$
Then the corresponding compact matrix is
$$ \begin{bmatrix} \boldsymbol{A}_1&\boldsymbol{0}&|&\boldsymbol{B}_1 & \boldsymbol{0} \\ \boldsymbol{0}&\boldsymbol{A}_2&|&\boldsymbol{0}&\boldsymbol{B}_2 \\ \hline \boldsymbol{C}_1&\boldsymbol{0}&|&\boldsymbol{D}_1 & \boldsymbol{0} \\ \boldsymbol{0}&\boldsymbol{C}_2&|&\boldsymbol{0}&\boldsymbol{D}_2 \\ \end{bmatrix} $$
But if you cascade them, the result is
$$ \begin{bmatrix} \boldsymbol{A}_1&\boldsymbol{0}&|&\boldsymbol{B}_1 \\ \boldsymbol{B}_2\boldsymbol{C}_1&\boldsymbol{A}_2&|&\boldsymbol{B}_2 \boldsymbol{D}_1 \\ \hline \boldsymbol{D}_2\boldsymbol{C}_1&\boldsymbol{C}_2&|& \boldsymbol{D}_2\boldsymbol{D}_1 \\ \end{bmatrix} $$
Are you following the parallel and cascade rules?
If hard to follow, you can simply use the traditional methods:
$$ \begin{cases} y_1=x_3-x_4+x_6+7_7 \\ y_2=x_2+x_5 \end{cases} $$
where
$$ \begin{cases} x_1=\frac1s u_1 \\ x_2=\frac1{s^2} u_1 =\frac1s x_1\\ x_3=\frac1{s^3} u_1 =\frac1s x_2\\ x_4=\frac1{s+1} u_1 \\ x_5=\frac1s u_2 \\ x_6=\frac1{s^2} u_2 =\frac1s x_5\\ x_7=\frac1{s+1} u_2 \end{cases} $$
Now easy to build state-space representation:
$$ \frac{d }{d t}\boldsymbol{x}= \frac{d }{d t} \begin{bmatrix} x_1\\x_2\\x_3\\x_4\\x_5\\x_6\\x_7 \end{bmatrix} = \begin{bmatrix} 0\\x_1\\x_2\\-x_4\\0\\x_5\\-x_7 \end{bmatrix} + \begin{bmatrix} u_1\\0\\u_1\\u_2\\0\\u_2 \end{bmatrix} $$
You can easily build $\boldsymbol{A}$ and $\boldsymbol{B}$ where
$$\boldsymbol{C}=\begin{bmatrix} 0&0&1&-1&0&1&1\\ 0&1&0&0&1&0&0 \end{bmatrix}$$
$$\boldsymbol{D}=\begin{bmatrix} 0&0\\ 0&0 \end{bmatrix}$$