I will follow the definition of the volume of a Riemmanian manifold from page 422 of Introduction to Smooth Manifolds
Let $M$ be an open and convex subset of $\mathbb{R}^n$ with non-empty interior and compact closure (e.g. $(0,1)\times(0,1)\subset\mathbb{R}^n$). Such $M$ clearly defines a smooth manifold, and we could take, for example, the standard Euclidean metric as Riemannian metric $g$. However, in order to define the volume of $M$ defined by $$ vol(M) = \int_M\omega_g, $$ $M$ itself must be compact. However, e.g. $(0,1)\times(0,1)$ itself is not compact. Moreover, the indicator function $f:=1_M$ is neither compactly supported nor continuous, hence $vol(M)=\int_M f\omega_g$ does not make sense.
Intuitively speaking, as $\bar{M}$ is compact, the volume of $M$ is bounded. However, how can we rigorously define the volume of such $M$? For example, how can we construct $\omega_g$ such that the volume of $(0,1)\times(0,1)$ is 1. Or do we need any additional condition on $M$?