In terms of public projects, Han China, like many contemporary and historical empires, sponsored a great number of public works. The first example that probably comes to mind would be the Great Wall of China. The Great Wall finds its origins centuries before the Han dynasty, but really only became a connected project under the Qin dynasty. The wall was expanded during the Han era, though what we know and think of as the Great Wall was really expanded and set up by the Ming. Obviously however, such a project was set up for defensive purposes and not for leisure.

In terms of the famous public projects focused on entertainment though, there don't seem to be many good analogs within the Han dynasty. There was the concept of the Hundred Events/Entertainments/Shows which included a lot of different types of events (as the name implies), mostly centered on martial arts (though they included singing, dancing, etc.). However, nothing written on the Hundred Events suggests to me that there were specific public work projects to support the events in the same vein as the Colosseum. Additionally, the Hundred Events were tied to displaying Han military power and wealth and were often done to impress foreign envoys, whereas from my understanding, comparable Roman activities were primarily done for the domestic population for the sake of keeping "idle masses entertained" (someone knowledgeable on Roman history feel free to correct me here, this is what I've always been taught though.)

However, if we are willing to look into Middle Imperial China instead, we can find many more comparable features. In Tang Dynasty China for example, we see the rise of public bathhouses (further popularised during the Song dynasty), the organisation and further development of early Chinese opera, etc.. It's during the Tang dynasty that we see the establishment of the jiaofang, which refers to institutions for musical education. Technically speaking, Tang China would be a contemporary of the Eastern Roman Empire, though I'm not too sure what their entertainment traditions were :)

Sources:

• Peter A. Lorge, Chinese Martial Arts: From Antiquity to the Twenty-First Century (2017)

• Charles Benn, China's Golden Age: Everyday Life in the Tang Dynasty (2004)