I'd have to see the picture in question to be sure, but it seems most likely that we're talking about hammock nettings. The hammock netting assembly consisted of a series of U-shaped iron stanchions connected by a coarse rope netting, forming a "trough" along the top of the ship's bulwarks. The crew's hammocks were stored in the nettings during the daytime; that arrangement, in addition to getting the hammocks out of the way, provided a barrier against musket shot.
In later years (after the middle of the nineteenth century or thereabouts), warships sometimes replaced their rope hammock nettings with genuine planked troughs. (The most famous example probably is the U.S.S. Constitution.)
Another possibility is that the painting shows the ship with "boarding nets" (moreaccurately labeled "anti-boarding nets") rigged. These pieces of gear were simply big, rope nets that were suspended over the ship's deck just before an action started, to make life difficult for enemy boarders. The Harriet Lane seems a bit modern for rope hammock nettings; I wonder if the painting in question shows her getting ready to repel boarders at Galveston (where she was captured by the Confederates).
Removing sails and stowing them belowdecks was not part of the normal routine of the sailing ship. At the end of a voyage, if the ship was expected to spend quite a bit of time in harbor, the sails would be dried out, usually by simply hanging them from the spars (or stays) when the ship was either at anchor or tied up to a pier. (There are quite a few photos of latter-day sailing ships drying sails - sitting stationary, with their sails hanging limp. But rolling up a piece of canvas in order to dry it doesn't work.) The sails would then be stowed away, either below decks or in some sort of building ashore. And there were cases (e.g., a voyage around Cape Horn) in which new sails, made of heavier canvas, would replace older, lighter, "fine weather sails" when bad weather was expected. In the case of a warship in service, though, the sails would normally remain attached to the spars (or stays, in the case of jibs and staysails) as long as the ship was in commission. Sails that were "set flying," such as studding sails and (before about 1800 or thereabouts) royals, would be furled to their yards and stowed in the tops.
Youth, talent, hard work, and enthusiasm are no match for old age and treachery.