Let all possible $2023$-degree real polynomials: $P(x)=x^{2023}+a_1x^{2022}+a_2x^{2021}+\cdots+a_{2022}x+a_{2023}$,
where $P(0)+P(1)=0$, and the polynomial has 2023 real roots $r_1, r_2,\cdots r_{2023}$ [not necessarily distinct] so that $0\leq r_1,r_2,\cdots r_{2023}\leq1$. What is the maximum value of $r_1 \cdot r_2 \cdots r_{2023}?$
1. **Achievability:**
   - Consider the polynomial \( P(x) = \prod_{i=1}^{2023} (x - r_i) \) where \( r_i = \frac{1}{2} \) for all \( i \). 
   - Then, \( P(x) = \left( x - \frac{1}{2} \right)^{2023} \).
   - Evaluating \( P(0) \) and \( P(1) \):
     \[
     P(0) = \left( 0 - \frac{1}{2} \right)^{2023} = \left( -\frac{1}{2} \right)^{2023} = -\frac{1}{2^{2023}}
     \]
     \[
     P(1) = \left( 1 - \frac{1}{2} \right)^{2023} = \left( \frac{1}{2} \right)^{2023} = \frac{1}{2^{2023}}
     \]
   - Therefore, \( P(0) + P(1) = -\frac{1}{2^{2023}} + \frac{1}{2^{2023}} = 0 \), satisfying the given condition.
   - The product of the roots is:
     \[
     r_1 \cdot r_2 \cdots r_{2023} = \left( \frac{1}{2} \right)^{2023} = 2^{-2023}
     \]

2. **Upper Bound:**
   - Let \( P(x) = \prod_{i=1}^{2023} (x - r_i) \) where \( 0 \leq r_i \leq 1 \) for all \( i \).
   - Given \( P(0) + P(1) = 0 \), we have:
     \[
     P(0) = (-1)^{2023} r_1 r_2 \cdots r_{2023} = -r_1 r_2 \cdots r_{2023}
     \]
     \[
     P(1) = (1 - r_1)(1 - r_2) \cdots (1 - r_{2023})
     \]
     \[
     P(0) + P(1) = -r_1 r_2 \cdots r_{2023} + (1 - r_1)(1 - r_2) \cdots (1 - r_{2023}) = 0
     \]
   - Therefore:
     \[
     (1 - r_1)(1 - r_2) \cdots (1 - r_{2023}) = r_1 r_2 \cdots r_{2023}
     \]
   - Let \( q_i = \frac{1}{r_i} - 1 \). Then \( 1 - r_i = \frac{q_i}{1 + q_i} \) and:
     \[
     (1 - r_1)(1 - r_2) \cdots (1 - r_{2023}) = \frac{q_1}{1 + q_1} \cdot \frac{q_2}{1 + q_2} \cdots \frac{q_{2023}}{1 + q_{2023}}
     \]
   - Since \( q_1 q_2 \cdots q_{2023} = 1 \), we have:
     \[
     r_1 r_2 \cdots r_{2023} = \left( \frac{q_1}{1 + q_1} \cdot \frac{q_2}{1 + q_2} \cdots \frac{q_{2023}}{1 + q_{2023}} \right)^{-1}
     \]
   - Using the AM-GM inequality:
     \[
     \frac{q_1}{1 + q_1} \cdot \frac{q_2}{1 + q_2} \cdots \frac{q_{2023}}{1 + q_{2023}} \leq \left( \frac{q_1 + q_2 + \cdots + q_{2023}}{2023} \right)^{2023}
     \]
   - Since \( q_1 q_2 \cdots q_{2023} = 1 \), by AM-GM:
     \[
     \frac{q_1 + q_2 + \cdots + q_{2023}}{2023} \geq 1
     \]
   - Therefore:
     \[
     r_1 r_2 \cdots r_{2023} \leq \left( \frac{2023}{2023} \right)^{2023} = 2^{-2023}
     \]

Conclusion:
\[
\boxed{2^{-2023}}
\]