*** UPDATE ***
I just received a very gracious note from Lily Chang, who is the Membership Services Director of the American Club.
She confirmed that the Club was originally located behind the Rainbow Hostel in the current Fine Art Museum area.
I was pretty sure that I had seen their tennis courts from the second floor of the hostel when I lived there.
Here are more great photos from Bill Thayer. This used to be the Club 63, which later became the China Seas Club when the Navy took over its management. Today the building is owned by The American Club in China (ACC), a private club that's mostly for international business people.
The ACC began in 1968 and moved to its present digs after the China Seas folded. I believe that it was located right next to the East Compound (just north of the hostels) when I was there in 1973-1974. I may be wrong about that (wouldn't be the first time on this blog) so if anyone knows for sure, please let me know. I've emailed them a couple of times, but they apparently don't respond to nonmembers.