Depends on what exactly you mean by space-time travel, i'll assume that you meant travel through both time and space by folding space-time.

 Space travel by folding or warping space has mathematically already been worked out by physicist Miguel Alcubierre. It involves creating a warp bubble that shrinks space ahead of the ship and expands space behind the ship, much like in star trek. Traveling through time would simply be a extension of this method into the fourth dimension.

However, the energy requirement would be huge, far greater than we could potentially produce with even nuclear power, also it would also require a singularity powerful enough to bend space to the degree we need it to. Currently we are limited to just barely being able to create subatomic singularities that exist only for milliseconds at a time.

In short you need

a colossal power source (antimatter, wich we could theoretically make with modern tech)

a way to make a controled and powerful singularity (again we can theoretically make this with modern tech)

exotic matter particularly matter with a negative mass would make creating the bubble much easier, but we havent found any yet.

So if you have $300 trillian burning a hole in your pocket, you could hire the companies to start making on and have it within a decade or two, there really are no barriers in terms of available technology, you just have to wait for the technology to become affordable.