Thanks for all your advice. So, I removed the Gypsy (I now know what a Gypsy is, and how to get it out) and took it to RV Boating And Marine, who tried some chains for size, and confirmed that it is a 6mm short link chain. So I got a replacement galvanised chain, see photo below - new chain on left, old chain on right, Gypsy in the middle.
On reflection, the biggest risk with the old chain is not that it would break at anchor. Although it is possible that the chain would break, there is a bigger risk that it would not run through the windlass when you need to drop anchor in an emergency. It would jam because the crud and rust could fuse a couple of links together and they would get stuck in the hawse pipes (which is what happened when I was removing the chain in the berth). Because I'm mostly day-sailing in benign waters (Moreton Bay), a broken chain probably means a lost anchor and some inconvenience to your schedule. Not being able to drop anchor because, say, you've lost your rudder (been there, done that), would be a bigger problem because you might end up with a grounding or a collision (thankfully, avoided in my case, but that's a different story).
I suppose I am justifying the new chain on the basis that it will run smoothly through the windlass. The old chain could carry considerable load for many years to come, but it will jam in the hawse pipes when I might need it most.
