The post of my rotary washing line has rusted into its sleeve in the ground and I do not have the strength or any powerful equipment to remove it. This happened over a period of a few winter months - does anyone have lightweight solutions that might do the trick? Would vinegar or oil help if I can pour them into the the tiny gap between the post and the sleeve? Thank you.
You actually have two problems, starting with how to get the post out of the sleeve now.
The first thing to try is some force, and the easiest form of force here would be a hammer. Circle the post, whacking it right above where it enters the sleeve (the harder the better, although use your judgement on avoiding damaging the post). Hopefully that will shear the rust and let you remove the post. Tapping down on the top of the post could also help (although it might also set it even harder in the sleeve).
Another possibility would be a strap wrench, which would let you twist hard on the round pole. You can find them for $5-$10 at discount hardware stores.
Vinegar wouldn't help, but a penetrating oil such as WD-40 might. However, you might need to use a lot and you don't want to create a teenie Superfund site in your back yard.
Your second problem is how to prevent this in the future. The obvious solution: don't leave the washing line outside in the winter! If that isn't possible or feasible, then you might consider replacing the (metal?) sleeve with a plastic one such as a PVC pipe. Bonus points for a bit of looseness in the fit between the post and the sleeve (although too loose and your washing line will look drunken).