In a picture, glue in a very small sphere via a much smaller small tube. The injectivity radius is now very small, as most geodesics on the sphere are still periodic, unaffected, because the tube is so small by comparison. The distances are almost unchanged, although it will take some work to make a proof, because it isn't easy to write out an explicit diffeomorphism between, for example, Euclidean space and Euclidean sphere with a very small sphere glued to it.

In a picture, glue in a very small sphere via a much smaller small tube attached at one end to the small sphere and at the other end to your manifold. The injectivity radius is now very small, as most geodesics on the sphere are still periodic, unaffected, because the tube is so small by comparison. The distances are almost unchanged. So for any Riemannian manifold, there is another Riemannian manifold with arbitrarily close metric which has injectivity radius arbitrarily close to zero. It will take some work to make a proof, because it isn't easy to write out an explicit diffeomorphism between, for example, Euclidean space and Euclidean sphere with a very small sphere glued to it.