I would do a couple of things to get a general direction on what methods may help. You may have to take more than one approach.
*** I know it may be obvious, but I have done a lot worse Make sure the AC is running when you do these tests ***
Your air handler may be amplifying noise, essentially being a sheet metal box drum. Sound coming from the metal also gets transmitted through ducting.
I would start by checking to see if the automotive style soundproofing material may help. Start by setting your hand on any of the uninsulated ducting sections or large metal panels. You don't have to push very hard. If the tone changes significantly or even drops in volume that is a good sign it can be treated. Dynamat is the name brand but there is also Amazon Basics Sound Deadening mat or whatever sound deadener you get at the best price. Cut out the shape with a utility knife and stick it. On one hand you may not have to cover everything, just some large strips. On the other you may need two or three layers.
Next is addressing sound coming through the door or walls. You can treat the walls inside the AC closet the same way as the door. In fact, they may nor ne insulated at all. If they are all interior walls. For the walls I would also think about blowing in Cellulose insulation if they are uninsulated and that's where sound is coming in from.
If sound is also coming through the door, mineral wool insulation is a great sound damper. Don't install it directly to the door. Know what thickness you are buying and make a frame. For example, if you are using 3.5" thick batt use cheap 1x4 lumber to make a frame the same profile as the door. Put the insulation in the frame and then use landscape fabric (its cheap) or other fabric that is very very breathable to cover the insulation front and back so it's held in place. A staple gun works great. Here is the key. Offset the frame so there is about a 2 3 in gap between the door and the insulation.
The alternative to this method for sound coming through the door is Harbor Freight moving blankets. Get about 5 or 6 is Harbor Freight moving blankets and hang them lengthwise covering as much of the door as possible. Space them out so there is about a 1-inch gap between the blankets and also the same air gap between the door and blanket #1.
One more thing - If your door is a hollow core door; pull the pins and take it to an actual no 'S' door supplier/installer not a big box store. Buy a solid core door, ask them to measure your existing door. You get to take you door slab back home with you and re-install it. When you pick up your new door it will be ready to go, door handle drilled out, hinges routered, etc. If it's a primed door you need to paint it. Otherwise slap on some hinges and a handle, pop in the pins and its done.
Good luck, reducing sound can be tough and often take more than one method to fix it. Also, if its straight up mechanical coupling through the ducting, framing, etc. You may not get a good return on investment and only minimal noise reduction. If that's the case or you want to try something with minimal labor and cost involved -- White Noise Generator or have you home assistant play nature sounds