While focusing on budget and technical issues is part of the discussion, remember that the aesthetics for film and television at the time were very different -- and that's probably more responsible for the look of the show than anything else. The "look" of Star Trek, with indoor sets, minimalist interiors, and atmospheric lighting is not inconsistent with many shows of the period, including non-sci-fi ones like Bonanza and The Man from UNCLE. "Mainstream" audiences by and large weren't interested in hyper-realism for what were what my grandfather used to say were "just made-up stories."
As further evidence, my mother and father, who were casual sci-fi watchers, along with others walked out of 2001: A Space Odyssey, in part because it bored them to tears and in part because its look was "cold and sterile." They expected something colorful and along the lines of what was normal then, a la Star Trek or Lost in Space but with a bigger scale -- instead, they got something that looked like a hospital surgery, and that was off-putting. A decade later the look may have been in vogue, but it wasn't in 1968.
And here's the best part -- almost everything that looks so modern and realistic today will look silly to audiences 40 years from now.