As someone who knows next to nothing about this space, why can China not build their own machines? Is ASML the only company making those machines? If so, why? Is it a matter of patents, or is the knowledge required for this so specialized only they've built it up?

They can - if they are willing to invest a a lot of money over several years. The US got Nuclear bombs in a few years during WWII with this thinking, and China (or anyone else) could too. This problem might be harder than a bomb, but the point remains, all it takes is a willingness to invest.

Of course the problem is we don't see what would be missed by doing this investment. If you put extra people into solving this problem that means less people curing cancer or whatever. (China has a lot of people, but not unlimited)

Yes. ASML is the only company making these machines. And both, they own thousands of patents and are also the only ones with the institutional knowledge required to build them anyway.