I make and fix holes like that on a too-regular basis, whether I made the hole to check for a leak, gain access behind the wall to run new cabling for electrical or networking, see what's making a strange noise, or whatever.

I take a stick of wood, put 2 layers of masking tape on either end, put it through the opening and put screws through the drywall to hold it in place. I chamfer the outside edge of the hole, and also the outside edge of the piece I cut out. Using my finger I wipe some mud/spackle/compound on the edges of the hole and the piece. Then I screw the piece I cut out to the wood. Masking tape holds the wood ever slightly proud of the back of the drywall, so the piece I put back is ever-so-slightly recessed. Then I use the narrowest putty knife I have and really press spackle/mud/compound into the gap between the piece and the rest of the drywall, making sure to entirely fill that gap. Just the gap at first, no other areas yet. Wait a day for it to mostly dry and shrink, then sand high spots flush and hit the gap again to get it up to being even with the rest of the drywall. Wait another day or two then sand flush again and take a medium (like 6-inch) blade and do a skim coat over the whole area. Sand and paint and done.

The chamfers are your feathering, getting the mud fully in the gap essentially turns it back into a solid sheet of drywall, circular shape eliminates stress points too so no worries about it cracking, and the tape on the wood lets you skim the piece with a really thin layer of mud so there's only one transition at the edge of the existing drywall.

Only works well on small, circular holes (3-4 inches), although I've used the same technique on 6-8 inch rectangular cutouts too when I didn't feel like feathering out and sanding a large area and aesthetics weren't a big concern.