$\mathbb R^+$ alone denotes the positive real numbers, and the subscript we see here $0$ denotes the inclusion of zero, as well. So all together, we have the set $$\mathbb R_0^+ = \{x\mid x\in \mathbb R, x\geq 0\}$$
This set is sometimes denoted by $\mathbb R_{\geq 0}$. There is no one universally used notation to describe the set of non-negative real numbers. So it's usually best that authors define the notation they plan to use.
The bar notation is often used in Lie Group theory.
For example, the $n \times n$ unitary matrices are defined as
$$U(n) := \{ X \in \mathrm{Mat}(n,\mathbb C): \overline{X}^{\top}\!X = E\}$$
where $\overline{X}^{\top}$ is the conjugate matrix of $X$ transposed, and $E$ is the identity matrix.
Just as the (real) orthogonal matrices $X^{\top}\!X=E$ preserve the (real) inner product
$$\langle u,v\rangle = u^{\top}v$$
the (complex) unitary matrices $\overline{X}^{\top}\!X = E$ preserve the (complex) Hermitian product
$$\langle u,v\rangle = \overline{u}^{\top}v$$
Could you tell us what $\mathbb{E}$ is, e.g. is it the identity matrix? If it is then I suspect that $\overline{X}$ might be a, potentially confusing, alternative to $\overline{X}^{\top}$. I've seen $\overline{X}^{\top}$ written as $X^*$ or $X^{\dagger}$ but never just $\overline{X}$.
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It's not an often-used convention, but in physics, matrices are sometimes appended with a double line underneath and vectors a single line underneath. This somewhat unifies the matrix/vector notation without the clumsiness of vector notation (and how to extend that to matrices).