Wikipedia has some interesting lists, including this one of a more general list of unsolved problems across many fields: https://en.wikipedia.org/wiki/Lists_of_unsolved_problems

Does anyone else feel inspired by reading this list? And fantasize about being able to dedicate multiple lifetimes purely to the pursuit of knowledge?

It is amusing in the late 1800s some physicists were declaring physics a dead end because most problems had been solved, save for a few rough edges.

There are people today making the same claim.

are there any unsolved problems in classical mechanics?

Some things should be probably updated on that page, I think the Sonoluminescence problem has been mostly explained.

The fact that there is an actual (short) list boggles my mind. (And yes I understand this is a case of "questions raise more questions" but still).

There are many open ended questions here. There's more flexibility to sci Fi then I thought.