A simple job but worth doing. Don't put up with too much 'wobble'. Up and down might be bearings, sideways might be shims.
Star hub rebuild. Remove, clean, and measure. Odd bits alone can be replaced but if it's all a bit tired you can replace everything with a complete kit. Oversize bearings are also available.
This kit is particularly good because this old hub didn't have a grease nipple. The cover on the new kit did so that's another advantage as its means you don't have to take the cover off to re-grease everything.
It's just two hours labour plus the parts. Bring your bike down and we'll do it while you wait. Don't keep putting it off.
Some black paint on that new chrome and a few road miles will soon restore that old patina!