ETR (einsteins theory of relativity) is based on the mathematics of the Lorentz contraction. That is, the faster you go, the shorter you become (length measured in the direction of travel gets shorter). As you approach the speed of light you become shorter until you reach a theoretical zero length at the speed of light.
As a result, Einstein defines the speed of light as being absolute, a maximum limit. Your thought experiment (repeated by millions including myself as a student) assumes particles and molecules (that make up a person) are travelling at the speed of light while walking around adding walking speed to the speed of light. Firstly, they aren't, and are travelling at a fraction of the speed of light. Even 'if' you could travel fast enough, the faster you go, the limit of the fastest particles within you would still only be the speed of light (in theory). So something is forced to change according to ETR theory.
Electrons in a copper wire travel a let than the speed of light for example.
FWIW: Much like the climate debate, ETR is not considered as gospel to everyone, and scientists were divided by claims of plagiarism and inaccuracy when ETR was first released in the early 1900's. Then after came Einsteins Special Theory of Relativity (a part 2).
(aside: most on this forum would know this well as disappearing up ones own rear end given how fast many travel to pull the trigger)
