How do I make the Wordpress media library be somewhere else?

I've been a bit short sighted. I set up my WordPress instance on a server with... Not a huge amount of space. A year or so later, and now I'm running out. I have another server which I could serve files from, and it's got loads of space. Is there a way that I could get the Wordpress media library to just... Serve and upload from that one instead? Preferably migrate it there as well.

Too many people I find online get some sort of yuppie answer like an expensive plug-in with an Amazon CDN subscription. I just want to homebrew this. I don't intend to host the major site assets for the site structure on it, and I'm fine with images taking a little longer to load.