Strictly speaking there's nothing wrong grammatically with this. What is unusual about it, and what makes it grate slightly to (at least this) native speaker, is that there is a clash of styles: "got" in this context is slightly informal (though entirely normal in conversation), whereas "most tired" is rather formal/old fashioned.
Either would be fine alone, in the right context, but putting the two together makes for a strange mix. To fix it, while keeping the meaning the same:
If you want to be formal, replacing "got" with "grew" is better than "became", which to me feels slightly clunky in this context.
If you want to go the other way use "very" in place of "most".
I also think it’s odd that there is only one leg. Unless it is known that the runner has only one leg I would think it more correct to say “my legs got...” or, if referring to a specific leg, “my left leg got...” – Fogmeister – 2017-12-03T09:48:03.547
1@Fogmeister: Some people do only have one leg! But also many people have one leg that's known to have issues (e.g. due to past injury); in those circumstances it would be quite normal to say "my leg got tired", implicitly referring to the injured one. – psmears – 2017-12-03T13:36:31.100
true. But you would have to know of predicting conditions in those cases... I think. Would be strange without that previous context. :-) – Fogmeister – 2017-12-03T19:19:19.493