If $(M,g)$ is a Riemannian manifold, it is possible to find an exhaustion $(U_i)_{i \ge 0}$ of $M$ with regular domains, i.e. with open relatively compact subsets having smooth boundary. Unfortunately, these domains may be unbounded in their intrinsic distances, i.e. in the distance produced by $g \big| _{U_i}$ (even though they are bounded in the Riemannian distance of $M$). (Think of $\mathbb R^2$ and take $U = (0,1)^2 \setminus C$ where $C$ is something like an intricated deleted double comb space, but with the segments of the comb being of constant length $3/4$ and thickness $2^{-n}$, with their thickness decreasing to the left: as a length-metric space $U$ is unbounded to the left.)
Is it possible to choose this exhaustion having the supplementary property that each $U_i$ is bounded in its intrinsic distance?
There exists $1$-net $p_i\in M$ i.e., for any $x\in M$, there is some $p_i$ s.t. $d_M(p_i,x)\leq 1$.
Define $V_j =\{ x\in M| d_M(x,p_j)\leq d_M(x,p_k)$ for all $k\neq j\}$, which is contained in a closed ball $B(p_j,1)$.
Define $U_1=V_1$ and $$U_i = \bigcup_{V_j\cap U_{i-1}\neq \emptyset}\ V_j$$
Since $V_j$ is locally finite covering, so we can make $U_i'$ with smooth boundary s.t. $d_H(U_i',U_i)\leq 1$ where $d_H$ is Hausdorff metric.
[Add] For $x_i\in U_i$, there is $p_j$ s.t. $d_M(p_j,x_i)\leq 1$.
In further, $V_j$ has intersection point with $x_{i-1}$ with $U_{i-1}$.
Hence by reindexing, $$ [x_ip_i]\cup [p_i x_{i-1}]\cup [x_{i-1}p_{i-1}]\cdots\cup[x_1p_1] $$ has a length $ 2i-1 $.
That is, any two points in $U_i$ has at most distance $4i-2$.