"leen to" 3 sides, and a roof. look at pole barns. to generic sheds.
on the lower 3 to 5 feet that is near the ground for the sides, i would most likely advise some sort of 2x6's or 2x8 lumber. cows, to horses, to sheep, to pigs, to other. they all will lay against and rub up against the sides and the lumber on the lower 3 to 5 feet. and instead of using metal or plastic sheeting a metal building or pole barn might use on the lower area. the wood. will hold up. ya there will eventually be some cracks between the 2x6's or 2x8's. on edge for the lower portions of the walls. but the lumber is acting more like a large wind block.
as far as width and height. make sure it is wide enough to get tractor in it, with FEL to remove all the manure and nasty straw, hey, etc... that will build up. you need to think about having enough room to not only drive staight into the area with fel bucket, but also coming in at some angle with the FEL to help clean stuff out. if you do a concrete pad, it can make life easier long term for you. when it comes to cleaning out the nasty stuff. ((note the lower lumber on 3 to 5 feet)) will see scares of using the FEL on the tractor from cleaning the area out. the height, it should be high enough to get the tractor into and under the leen to. and have a foot plus of clearance, so when you need to back drag some stuff and go over a pile of stuff with tires you do not run tractor into the ceiling.
a lean to has 3 sides and a roof. so locating it, so wind is kept down to a min on the open side is suggested. and positioned in a way so the wind go around it. vs directly into it. you want things large enough to allow the live stock (cows, horses, etc...) to be able to pack in, (stand side by side) and also lay down, and keep the wind off of them, along with rain and/or snow drifts from building up in the area were they have a dry spot to stand / law down.
the upper portion of sides and roof, metal sheeting can be used. i would advise use of gutters. and good drainage from the area for regular run off water. poo to uruine, to other. will build up over time. and keep run off water from the roof to other areas around the leen to. will be needed. just adding rain water / snow melting to poo and urine can and will only make a bigger mess. and if poo / urine drains into a near by creak or lake, or pond, or is to close to a well, you could be in big trouble from fines, to a lot more extra cost of fixing issues.
if you need a fairly high clearance for the tractor. you can use "tarps" for upper area on open side, to help close some of it off. during winter. leaving the lower potion open for animals to go in and out freely without nocking into the tarps.
you will most likely want to build things "large" enough in order to put in some sort of grain feeder or hay feeder. along with area to stack straw bails out of area were animals will be, for use as a beding for the animals as needed. along with placing a live stock water tank that is out of the extra hot and cold weather temps of the year.
if you look up local "agriculture" website for your state, or simple contact your local ag department for your county and/or state. they might be able to provide a few extra things.
would advise looking at "pole barns" here on the forum. along with going to the old local library. the books will be old and dusty, but not much has changed from long ago in how to build a barn / shed / live stock area.
