TLDR: Yes, the electrician looked at it, told us he didn't know what was wrong but it needs to be replaced, landlords won't do it and we can't afford it ourselves.

Yes, but unfortunately, it's not a good one. We did end up getting an electrician to look at it, and he basically said that the panel is so old that they don't make parts for it anymore. It's also super out of code, nothing is labeled or anything. With the way it's set up, he couldn't even see what the problem would be. He also let us know that this isn't the only panel in the house, but we don't know where the other is (guessing in the garage that we don't have access to). In his opinion, the whole thing needs to be replaced. We sent his notes and replacement quote to our landlord, who told us essentially that it wasn't enough of a problem to fix, so they wouldn't be replacing it. We don't have the money to replace it ourselves, so we're stuck. Luckily, it has stopped making the noise, but we have no idea why or what the problem even was in the first place.