Unfortunately the version of a Dyson sphere depicted in both is completely untenable, according to Freeman Dyson himself. He reportedly enjoyed the episode as "a piece of cinema" but denounced the science involved as 100% fiction and not possible. As others have said, perhaps building something like that around a dwarf star would be within the realm of possibility, but you still have issues such as radiation, constant daylight, gravity, and drift (the sphere would eventually drift into the star since it's not actually orbiting and is much more massive than even a large planet).
Still, it's a really cool idea and a fun thought experiment.