A random symmetric matrix

Idea. Condition on the diagonal. Both diagonal entries must be nonnegative, and then the off-diagonal entry must be small enough. Each condition is a simple probability.

Step 1: the diagonal. $a\ge 0$ and $c\ge 0$ with probability $\frac{1}{4}$. Given that, $a$ and $c$ are uniform on $[0,1]$.

Step 2: the off-diagonal. Given $a$ and $c$, $b$ is uniform on $[-1,1]$, and $|b|\le \sqrt{ac}$ has probability $\sqrt{ac}$ (which is at most 1).

Step 3: average. $\mathbb{E}[\sqrt{ac}]=\mathbb{E}[\sqrt{a}]\mathbb{E}[\sqrt{c}]$, and for $a$ uniform on $[0,1]$, $\mathbb{E}[\sqrt{a}]={\int }_0^1\sqrt{a}da=\frac{2}{3}$. So $$\mathbb{P}=\frac{1}{4}\cdot \frac{2}{3}\cdot \frac{2}{3}=\frac{1}{9}.$$

Remark. With independent standard normal entries instead, there is no simple closed form; the probability is about $0.116$.