What they can't build is a gas turbine that is competitive with the existing engines from companies that have been developing them for decades.
China has a strategic interest in developing military gas turbine engines. Right now, they are totally dependent on Russia. If the two nations have a falling out, as they have in the past, then the PLA's air force could be in some amount of jeopardy if their primary source of engines is cut off.
Japan and Germany have no real reason to get into the aircraft engine market, although Japan does manufacture smaller gas turbines under license from RR, P&W, and GE. Both nations are on good terms with multiple engine suppliers in multiple countries, so their military engine source is very unlikely to be cut off due to political reasons. As for commercial gas turbines, neither nation could justify the considerable funds required to field a competitive aircraft engine.
The gas turbine market is fairly mature, and fiercely competitive. To build a turbofan that can rival the existing products in size, weight, capacity, and especially fuel usage requires the cumulative experience of the last 70 years. Such an effort needs the sort of knowledge that you just can't pick up in an internet search, because the organizations that make the best engines guard their designs very closely.
As the market is both mature and very competitive, the profit margins are not so high that a new company could be set up to make gas turbine engines and expect to show enough profit to offset the considerable expense. It would take decades of expensive development just to try to match what the current producers make, and they aren't just sitting on their laurels. P&W is trying to work out the kinks in a geared turbofan that promises to be more efficient than current ones, and GE just made the first flights with the GE9X, a huge engine to power the 777X that's due out in a few years.
This competitiveness also holds true for military engines, as a better power to weight ratio means a better performing aircraft, while a more efficient engine means greater range and less refueling. If you want to build your own military gas turbines, you had best be sure they're in the vicinity of a possible opponent's engines, or you'll be at a distinct disadvantage. That homegrown engine isn't very useful if the planes it powers keep getting shot down due to inferior performance.
So, unless a nation has a strategic reason to get into this very difficult field of engineering (as China does), it's just not worth the expense when you can buy state of the art engines already fully developed, from a reliable source.