Frais du Bois (woodland starwberries), and tiny , thimble-sized wild strawberries. I have only taste them twice, usually they are eaten by bitds and squirrels…they grow in the lawn. Can you grow these things commercially?
I don’t have a direct answer to your question, but I bookmarked this article a few years ago, and intend to try it once I have a yard. It should help you get started! Let me know how it goes. 
There’s a reason the commercial strawberries you buy at the grocery store are large and rock hard. Large hard fruit are easy to pick, transport, and store. Tiny soft fruit are hard to pick, transport, and store. You might be able to find speciality berries at a farmer’s market, but be prepared to pay through the nose. The best way to get top quality strawberries is to grow them yourself.
If you’re out looking for wild woodland strawberry plants, don’t confuse those with the mock strawberrycommonly found in gardens and lawns.
Yeah, I was about to post that the “strawberries” in my yard don’t taste like anything, plus they grow too low to mow, the sneaky bastards!
In my experience, home-grown, organic strawberries are very small, though perhaps not as small as those you describe. The way you get the typical huge, flavorless ones is by using a lot of pesticides and unhealthy soil.
I wouldn’t think they’d be practical commercially, for reasons stated above. We’ve got them in our back yard–in four years now, I’ve never had a taste. 
The birds, squirrels or bunnies get to them first.
However, this year I finally remembered to make up some pots and try and capture some of the runners and see if I can get some started in a container, which I could move someplace where the birds, squirrels or whatevers won’t get. The runners are quite numerous, and the strawberries themselves seem to thrive in rather inhospitable, gravelly, oft-trod-on areas in the back, so they’re tough little buggers.
I remember lying on my belly in the woods with my friend Sara, eating wild strawberries, the size of our fingernails at the largest, in the summertime when I was about 14. They were so good, I still remember that afternoon, almost thirty years later!
I had a few and they taste like … well… a little acidic but mainly nothing. They might be a bit stawberryesque, but it must have taken quite few breeding generations to get a powerful strawberry flavor out of the little devils.
In looking at this article I might be wrong as it appears I was tasting them before they were mature.
Like Savannah said, the taste of wild strawberries is not something you’d forget. I remember hitchhiking to Sweden when I was nineteen. My friend and me spent a lot of time waiting by the side of woodland roads, and we ate wild strawberries, wild raspberries and wild blueberies untill we nearly burst.
Good times.
Wild strawberries should have a very aromatic strawberry flavour - when picked at the peak of ripeness, they taste and smell so intense, it’s almost hard to believe the flavour is natural.
They’re not usually very productive plants in terms of fruit per square yard though - so it’s unusual to find them offered for sale commercially - also, as Lemur866 says, they’re small and soft (and they don’t keep well) - all of which does not lend them to appearing on sale in the shops.
You can buy packets of ‘alpine strawberry’ seeds - wild and alpine varieties of strawberry are commonly grown from seed, unlike commercial dessert strawberries, which are usually propagated vegetatively in order to keep the variety true.
In a tower-type strawberry container, a few dozen plants, properly fed and watered, may produce a small handful of fruit every couple of days at the peak of their season.
I use them as ground cover in my garden, which is a bit shady and has a bit of a woodland theme to it.
A little more info on true wild strawberries (in England):
http://www.atomicshrimp.com/st/content/wild_strawberries