What's this setup on the side of my house actually for?

We bought a place a few years back with almost the identical setup on the rear wall, and it drove us nuts for months. Turned out the disconnect fed a line buried out to where a hot tub used to sit. You could still see the outline of the spa pad in the grass and a capped conduit stub sticking up near the patio. Walk the yard and look for anything like that, a concrete slab, a patch of gravel, or a capped pipe near where you'd put a tub or a shed. Also check the panel inside, if there's a big two-pole breaker labeled for somethig like that, it'll tell you the amperage the circuit was built for. A hot tub ussually gets its own double breaker, a plain outdoor outlet wouldn't.

One other thing, a lot of those red emergency disconnect labels are just code now. Utilities and inspectors have been requiring a shutoff outside so firefighters can kill the power without going inside. So don't read too much into the label itself. The tell is the wire size and where the conduit goes, not the sticker.

On my old place the setup was almost the reverse of yours, and the answer turned out to be mundane. The disconnect on the siding fed a subpanel in the detached garage, and the breaker panel in the house was nowhere near it. The electrician had labeled the outside box 'emergency disconnect' becasue that was the shut-off for everythign out back, not because it fed anything exotic. So the label alone doesn't mean much, they slap those on for code reasons even on boring circuits.

Before assuming hot tub or RV, I'd pull the cover off the panel in your garage, just look, don't touch anything, and read the breaker labels. If there's a big two-pole breaker marked for something like a shed, well pump, or pool, that's your answer right there. Mine said 'shop' in sharpie from 1987 and that settled it. Also check wheather the conductors going up into that eave keep going somewhere, sometimes you can follow conduit or cable along the soffit toward a light or an outlet on the other side of the house.

One more thought on the outlet location. Rear-of-house receptacles were pretty common just for yard gear, Christmas lights, or a grill rotisserie, especally when the garage wall was the easiest place to tap power. The heavy conduit might be oversized because the same run was planned to carry more later, or just because the electrician had a spool of it on the truck. If the family left any paperwork, an old inspection sticker or a receipt in the closing files sometimes lists the permits, and those usually say exactly what the circuit was for.

Mine was almost the same story but the answer was an old well pump. The disconnect was still live long after the pump got abandoned, just sitting there feeding nothing. If you flip the biggest double breaker in the garage panel and the outside outlet dies with it, you've got your answer they're one circuit. Also worth asking the closest neighbors, ours remembered the guy who owned the place before and cleared up two mysteries in five minutes. Old-timers on the street know more about a house than any paperwork does.

Check your county's permit records online, that's how I figured out a mystery box at my last place. Turns out the old owner pulled a permit for a shed with power in 2011, never finished it, and the disconnect just stayed. Most assessor sites let you search by address for free and it takes five minutes.

Also worth checking if the listing photos from when the house sold earlier show the backyard. Old pictures sometimes catch stuff that's gone now, like a trampoline, above ground pool, or whatever used to be plugged into that outlet.

Update: took the permit records tip and searched my county's site. Sure enough there was a 2014 electrical permit for a shed circuit, never closed out. The disconnect feeds conduit running toward the back fence where the old owner apparantly planned to build. Outlet does cut off with the big double breaker, so it's all one dead-end run. Mystery solved.