Get some steel wool and pour either lime juice or lemon juice on it and start scrubbing!
Hmmm, I would have said get a bag of steel wool and some rubber gloves....cover your nose with a bandana or a cloth...protect your eyes and rub away!
The cheapest and easist way I have used to remove rust is to use molasses. Get some from a farm or pet supply store (I think horses eat it) mix with water at about 10 or 15 to 1 (10 parts water, 1part molasses.) Then submerge the rusty item in the liquid ( you might need a large tub). After a few days the rust will become grey sludge and can be rinsed/scrubbed off. I don't know how it will go on the parts that still have some paint on them, and you may need to remove any cloth type coverings, but it is definitely cheap and easy, and can't hurt to try. It's also safe to use and won't remove any metal or other material except rust. I believe it also works on oxidised aluminium surfaces.
sandpaper navel jelly sandblasting electrolysis wire brushing with a drill
9" angle grinder ?
I'd add "rust conversion primer", available from auto parts stores, which I've become fond of. Unfortunately it's black, so unless you plan on painting over it... Speaking of painting over things: Are the chairs iron and rusting, or are they aluminum or wood with rusting fasteners?
Eastwood rust solutions.