Because I want cheap and flavorful produce.
Heck yeah. I love gardening and getting down and dirty but I also have a planter area set up that fits what you’re trying to do pretty closely. For me it’s my plant hospital that I’ve set up in the side yard where I do my potting, triage and recovery. I stacked two columns of plastic crates and put a marble slab on top. This way I work comfortably, can store stuff like potting soil underneath so it’ll be protected from the rain and I’ve included a series of shelves where I keep specimens until they’re ready for the patio, replanting, whatever. There’s no reason those couldn’t be left there indefinately. For convenience, watering and clean-up, make sure you set this up near a faucet. You’ll find a wide variety of herbs, vegetables and the like from a competent nursery. It’s really nice to be cooking and just fresh pick whatever you need.
Can’t find a farmer’s market? Buy from a neighbor?
Got any advice on bench gardening or indoor gardening or hydroponics? There seems to be a lot of good advice coming from other posters.
My mom mastered the art of gardening without bending, kneeling in the dirt or sweating. In fact, she was able to garden* without leaving the house and only using her voice. It just requires that one acquire that all-purpose piece of equipment known as a son.
*She was always telling folks how much gardening she was doing.
Sure, but nothing new.
Just about anything can be grown in containers, if you use a container suitable for the plant. Plants will only grow as much as their root system can expand to, so re-potting is often necessary. For someone who despises dirt and doesn’t want to touch it, I fail to see how container gardening will work out.
I once grew carrots in an abandoned kitty litter box lid. It was just deep enough to give the carrots somewhere to grow to. In my experience, if there are holes in the bottom for drainage, you can grow something in it. They also have those hanging planters that can be suspended from on high, so no bending or stooping to reach for watering purposes. You can even get those containers to grow tomatoes upside down; I haven’t tried those.
As previously mentioned, you can place all containers up on a rack or a potting bench or some sort of table. I always use some sort of water preservative that helps hold water in the pots because it tends to run through and the plants dry out fast. Some veggies like moist soil so would either require watering every day or some sort of water-retaining medium mixed in with the potting soil. This is a good re-use of shipping peanuts.
But yeah, all of that requires touching the icky soil, so… good luck with that.
He said “cheap.” 
I used to be an avid gardener. Loved to get down and dirty with no qualms about contact with the earth. But that was before severe knee problems set in. I can no longer kneel . . . or even worse, squat. So most of my gardening takes place in my enclosed patio.
Meanwhile, morning glory vines are taking over the yard. Thousands of them, working their way toward the house. Nothing kills them. I wish someone would make a weed killer that only attacks the morning glories, not everything else.
By the time I get my knees fixed and can return to the yard, nothing else will be left.
Why not get all feudal and allow serfs to work your land. Take a percentage of what they produce.
A gardening bench with containers is workable, though it may be a bit more fiddly than a garden bed–you have to re-pot things from time to time and pay more attention to watering, among other things. Hydroponics is even more fiddly, but it gets you completely away from the dirt. Both are easier to keep weed-free than a bed.
If your major objection is the bending and kneeling parts, a keyhole garden might suit you. It’s a small, round raised bed with a notch in it, so you can reach the whole area without too much bending. It’s also designed to take advantage of composting kitchen waste for both fertilize and water, which is particularly useful in relatively dry areas.
Keyhole garden was exactly what I was going to suggest as well.
Serf sup?
I do sympathise, both with those with knee or hip problems, and those who just don’t like being in grubby nature.
If you have a deck with railings, you could try a deck railing planter (here is one example, just for the visual - I’m not recommending this particular one especially). Just google “deck railing planter” and you’ll see lots of them. They allow you to garden while standing on your nice clean deck.
Roddy
Very nice.
On one of those HGTV shows they propped an old wooden pallet against a wall and stuck filled pots between the slats. Plus you could train some beans up part of it (runners); we grew some snow peas that way once. And tomatoes are grown “up,” in cages, so those aren’t too messy either.
Container gardening can yield flavorful produce, but if you’re planning on using bagged soil it may not be very cheap. And you’ll have to be more diligent about watering than you would if you we’re growing your plants in the ground, as container plants dry out faster. But it can be a lot of fun.
Am I the only person reading this thread who’s wondering how the OP and his wife are going to handle their first encounter with a tomato hornworm? 
By all accounts I’ve always hated getting my hands dirty, would freak out if I got anything sticky on my hands, and never made mud pies as a toddler etc. I do garden, though. I have lots and lots of gardening gloves, and those foam pads you’re meant to kneel on while gardening. They don’t keep 100% of the dirt off, but a fair amount of it.
Every gardener should have at least one good pair of gardening gloves! Not only do they make gardening a much cleaner experience, they make it a less painful one as well. (Especially when it’s time to prune the rosé bushes…).