In 1890’s H.G. Wells famously takes a “science fiction” approach to time travel by building a machine that can move freely through time as a dimension. It was made of clockwork — mechanical gears and such. Gears can perform ordinary motion in precise ways, but nothing about that enables doing something that normal motion doesn't.
Later treatments use whatever the technology of the day may be: electronics and then cybernetics. But again, that’s completely implausible, since electomagnetism experiences time in the normal way just as gears do, and calculating fast is supposed to do what exactly?
Now, we really do know about some “hard” S-F approaches, but these require giga-scale engineering and exotic ingrediants like negative energy.
I'm not talking about the projects of a K-Ⅲ civilization, though, or even a national scale project like a major particle accelerator. I'm wondering how a small scale effort, like a mad scientist or university professor, could (more) plausibly build a time machine. To clarify, he uses the machine to go back in time. (Forward is easy!) The setting is “near future” (like 15 years or so).
You have to consider why it hasn't been done already if it’s doable using ordinary stuff involving current technology. Technology based on new principles only just becoming available must have some plausible reason for allowing time travel, and not just be something new and mysterious. I won’t rule out distinguished artifacts per se, but I want the solution to be something that anyone could have done — finding the crashed UFO in your own basement is simply cheating.
The science-based explicitly rules out pseudoscience, but we allow generous handwaving. This kind of handwaving, done well, is exactly what would make for “plausible” storytelling. Specifically, the movie Primer was delightful in using an effect that I'd read about, which provides for interaction between gravity and electromagnetism via a superconductor. This was in New Scientist, and the theory is crazy pseudoscience. For someone who hasn’t looked into it, though, it comes across as plausible if you don’t push the details.