The main problem is that to push a macro-scale object (like a human, and enough equipment to keep them alive) through a "closed time-like loop/curve" that would allow going back in time would either require that you find a rapidly spinning black hole, and one that's big enough so that the gravitational gradient/tidal-forces don't rip you apart.
And it would require a spaceship capable of going at least a few tens of thousands of light years to go find such a black hole, if there's one in the Milky Way that would suffice. (If not, next stop Andromeda... about 2 million light years away...) And then the space ship would have to be massively armored so you can survive the Hawking radiation and accretion disk radiation and heat/and particles-gasses moving at possibly near light speed around the black hole.
If you and the space ship could survive all that, I suppose "over-shooting" your time travel destination by umpteen thousand years in the past so you could travel back to Earth and arrive back in 1950-whatever is trivial.
The other possible method I've heard, assuming you can't find the black hole of sufficient size and spin, is to make your own wormhole, using star or planet-sized quantities of mass/energy focused onto a subatomic quantum scale rip open a wormhole, keep on pouring mass-energy into it to keep it open. (similar issues to surviving the black hole trip, gravitational gradients and radiation that'll melt anything...)
And then, while keeping the wormhole open... you have to somehow drag it's mouth around in a circle near the speed of light, while you're entering it.
The one problem I see with this is that these experiments, or mathematical thought experiments presume that time actually exists, that it's a "thing" or a real property of the Universe. There may not actually be "space-time", but just "space", and removing the time element, all of physics, and even relativity, special and general, it still all works.
Everything may just exist in one "long now" where matter and energy is free to move, and what we perceive as "time" is just that entropy never... almost ever ever ever goes "backwards" You fill a jar with two layers of white and black sand and start shaking it, in theory, it could go back to two neat layers, but statistically it'll take trillions of shakes. The two layers neatly separated is what we perceive as "before" and the gray salt-n-pepper mess of mixed sand that will likely never go back to "before" is "later".
