2 adjusting to an edited question; [made Community Wiki]

If I understand you right, you're assuming that there is already a Riemann metric chosen for you. This of course integrates to a distance function ("metric" in the sense of metric spaces) whether or not the manifold is complete. Then one can do the usual thing of freely forming limits of Cauchy sequences which must be equal for any two sequences that are mutually tethered. Then one has to ask whether the completed space is a manifold. Of course it might not be. For example, take any orbifold you like that isn't a manifold, put a sensible Riemannian metric on it by unfolding singularities and averaging in an equivariant way --- and then the usual partition of unity lets you glue the bits together just as you'd like. Then the complement of the singular subspace is a manifold, and of course it doesn't sit in a complete manifold, because its completion really is what you started with --- so it sits an a complete orbifold.

Worse examples could be constructed, but that's the general idea.

Edits to the original question suggest that what's really sought is is something like a smooth extrapolation of a Riemannian metric, and sufficient conditions on the manifold to get such a thing. I believe the orbifold examples are still "bad" for this purpose; one can also build spaces such that the intrinsic diameter of the boundary diverges, but still have finite diameter on the whole. This can come in a variety of shapes --- for instance, the Koch snowflake is a flat example and clearly makes a nice open subset of the plane. One can also build variants with unbounded curvature but finite (gross) diameter --- here the unbounded curvature will be the obstruction to smooth extension... (more to come. I'm cw/ing this answer now.)

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If I understand you right, you're assuming that there is already a Riemann metric chosen for you. This of course integrates to a distance function ("metric" in the sense of metric spaces) whether or not the manifold is complete. Then one can do the usual thing of freely forming limits of Cauchy sequences which must be equal for any two sequences that are mutually tethered. Then one has to ask whether the completed space is a manifold. Of course it might not be. For example, take any orbifold you like that isn't a manifold, put a sensible Riemannian metric on it by unfolding singularities and averaging in an equivariant way --- and then the usual partition of unity lets you glue the bits together just as you'd like. Then the complement of the singular subspace is a manifold, and of course it doesn't sit in a complete manifold, because its completion really is what you started with --- so it sits an a complete orbifold.

Worse examples could be constructed, but that's the general idea.