How are you calling that external service? If it's truly a base url you might be better of using "named credential" for it. It'll abstract the base url away for you, include authentication or certificate if you have to present any...
Failing that - custom metadata might be a poor choice. They're kind of dictionary objects, you can add more (but not from apex) but if you deploy stuff using Git/Ant/SFDX CLI rather than changesets it'd become bit pain, you'd need different custom metadata value for sandbox vs prod. Kinda defeats the purpose.
You might be better off using custom setting instead (hierarchy is enabled by default, list you'd have to flip a checkbox in setup. List is useful if you need key-value kind of pairs, similar to custom metadata): https://salesforce.stackexchange.com/questions/74049/what-is-the-difference-between-custom-settings-and-custom-metadata-types
And you can modify them with Apex too. Which means that in ideal world you could have a "postcopy" class running as soon as sandbox is refreshed that overwrites the custom setting with the non-prod value. For named credential I don't think you can pull it off, you'd need a mini deployment that changes it or manual step (have you seen https://salesforce.stackexchange.com/q/955/799 ?)