"Oh, what sweet agony the preservation of antiquity". First, butyl caulk the seams, cracks and other leak points - and don't spare the butyl. Then soak the inside surfaces with gobs of oil Chevron Shingle and Floor Oil). All this assumes the insides of your gutters are unpainted.

If paint exists, the job gets harder. First scrape and/or wire brush away any loose paint, and thoroughly rough up all other painted surfaces. Then use a paintable silicone caulk at all joints. Next, apply and oil-base primer and an oil-base finish coat.

Wood gutters also can be lined with galvanized sheetmetal or copper, but either is very expensive.