No, the condensor unit cannot be enclosed. Airflow must not be obstructed.

Is the noise coming from the compressor or the fan motor? If from the compressor, it may have a broken internal support which is causing excess vibration. Have you called a servicer to investigate the situation?

Here are is YouTube video for some ideas:8 BEST Ways To Soundproof NOISY AC Unit!

Installing a sound blanket will help. There are videos showing this as a DIY project, but unless you know what you are doing it best to call a HVAC company to do a preventative maintenance service check and install the blanket at the same time. The tech can also check if the noise is normal or it is a sign of problems.

You can build a solid enclosure but you have to allow for adequate clearance as per the manufacturer's instructions on all sides and make it easy for future servicing. You could also build a type that allows air flow if the space is tight, but that won't be as effective for noise reduction.

If the noise is coming from the compressor, a sound blanket may reduce some but this is typically pretty minimal. If the sound is coming from the fan there's not a lot you can do about that either - make sure the fan blades are balanced and aligned (bent) correctly.

@dadoes No we haven’t had anyone look at it yet — I just wanted to get an idea of what we might be looking at. There is a ton of room where we could allow clearance around a fence — and idea how much space might be adequate?

@mike_home how much clearance do you think is reasonable if we built an enclosure? There’s room for a fairly large one.

I recommend at least two feet of clearance in front and on each side.

I picked the Carrier 24SCA4 air conditioner as an example. Page 11 of the Product Data shows some clearance examples. You could do the same check for your model.

Sometimes better advice requires better information and understanding. I could give you a laundry list of ways to dampen noise but most are specific to the type noise and most important the source. Only experience allows one to narrow cause and/or source down soooooo you might think about posting a clear recording of noise in hopes it being recognized by someone. Hint,start 20-30 feet away and close gap then reverse away again. Meanwhile press firmly on housing in various places with heel of your hand to see if certain spots quit the noise partly or completely..