We are trying to make our side yard look aesthetically pleasing. The HVAC compressor cage is an eyesore, as shown in pic below (the gutter were only temporarily removed for painting). What can we do to hide/make it look better?
I'm sorry to be so blunt but the A/C isn't making your yard look bad, your yard is making your yard look bad. That wood barricade needs to go!
Search the internet for "landscaping around A/C unit" and you should get plenty of inspiration like the image below:
The main thing to keep in mind is that the unit needs a minimum 2 feet of clearance around all sides so that it is easily serviceable and replaceable when the time comes. A simple rock surround usually does wonders for A/C aesthetic.
If you decide to plant shrubs then just remember that the coil fins will clog with debris often so you'll need it cleaned once or twice per year.
If you truly just despise the look of HVAC compressors then you could camouflage it:
As a HVAC service professional I wouldn't box the condenser with anything, this is problematic in multiple ways. If you want the system to remain problem free you must consider the service it will receive throughout its service life. With that being said, the 2 foot or 18" rule also would not suffice. I would recommend a single fence 4' from the condenser runing along the wall enough to block view. You could use a hedge row but not anything that will grow wild into the unit if left unpruned. In order to get the best service to the condenser you need to make it readily accessible.
Remove the cladding. For a side yard the compressor units are not so ugly. They're hidden from the back by the wider rear of the house. You can put a small attractive privacy panel facing the front, if these can be seen from the road, to make that view more attractive. The side view doesn't matter. The cage is doing more harm than good, and IMO the more you do the worse it will get as you draw more attention to it. Peoples' eyes are used to seeing those units in a side yard.
Also, with no cage, the area around the compressors is easier to keep clean. Otherwise, the view from above (out your side windows) will be down to a cage filled with leaves and crap unless you make special effort to clean it regularly.
I would suggest planting shrubbery around it, it will help absorb the sound (your neighbors will like that). The shade will help the unit keep cooler. Keep the shrubbery (assume it size in a few years) far enough away not to block the air flow and allow access to the unit for service. If you use evergreen type of shrubs it will be hidden year round. You can tie this in with the rest of the landscaping and it will look great and the AC will be much harder to see.
Whatever you choose, remember that this is where the heat from your home exhausts. Make sure that air is easily drawn in along the sides and can easily escape the top.
Some kinds of sheds would obstruct air flow. Even excessive bushes can create a problem. If it cannot exhaust the heat, the life of your AC will be short.
There are real reasons these items don't come in enclosed sheds. Perhaps you can build a surround that has wood angled such that it is not a real wall, but just a few pieces of wood blocking the view from above, and to a great degree, the sides.
If you build an enclosure, remember that you'll have to remove the AC, so the enclosure will have a gate. You'll have to occasionally spray the fins of the coil with water to clean the outside unit, so the enclosure should permit such cleaning. Finally, anything that covers the top from an airflow perspective will just stop the unit from working by blocking the heat after the exhaust.
I agree with Solar Mike.
Build a shed that matches the building and paint it the same. Extend a wall of the shed so it blocks view of the unit, i.e. so it appears from the outside to be part of the shed but in fact the non-obvious walls don't exist. Then, put all the crud in your backyard into the shed, and have a moratorium against new crud.