I recently got my hands on an old Olympia club machine that my parents have had since 1981. I was told "it leaks" and "maybe needs a new gasket". I'd love to repair it, however as I read through other posts, I realize I may be in over my head. Is there any way to walk me through diagnosing the issue, so I can figure out where to go from there?

Very cool!
Others may chime along with more directed advice, but I'd probably start with the Orphan Espresso YouTube channel. While most famous for work on the Cremina, I'm sure the series would provide some use.

Good luck!

I restored an Olympia club in 2018. Amazing little machine built like a tank!
They are not that difficult to restore.

Here are a few links:
Olympia Club Restoration DIY Rebuild

Ongoing Olympia Express Club Rebuild

1976 Olympia Club Rebuild

Cerini in NY sell olympia parts (stainless steel replacement screws are a must) but I would contact Migg https://olympia-kaffeemachinen.ch
for gaskets, spring and what not...

If I were you, I'd simply go through a full rebuild. Nothing wrong in changing 38 year's old seals...

She looks good btw! Good luck!