I've been admiring (and tasting) this interesting plant --- one of several Jim couldn't resist when selecting seed for this year's Heirloom Garden. We're calling it Strawberry Spinach, although it also is known (among other names) as Strawberry Blite, Strawberry Goosefoot, Indian Paint and Indian Ink.
Yes, both the leaves and berries are edible. I've eaten both straight from the plant, just out of curiosity, but the leaves would work as part of a salad mix or, reportedly, cooked as greens. The berries, more like raspberries than strawberries, are faintly sweet but have no particular flavor. The general recommendation is to use them as garnishes.
The plant has a number of Latin names, too --- generally considered synonymous: Blitum capitatum and Chenopodium capitatum, the most common.
Seedsavers Exchange identifies the plant as something "grown for centuries in Europe," and I did find a few references to monastery gardens and those sorts of romantic places.
Much of the time, however, it is identified as a plant native to North America, spread widely across the northern and southwest United States and Canada. Native Americans reportedly used the juice of its berries as dye, hence "Indian Paint" and "Indian Ink."
It would be interesting to see what homespun yarn infused with Strawberry Spinach berry juice ended up looking like, but someone else will have to give that a try.
Anyhow, if you're interested in taking a look for yourself, feel free to walk down the hill to the Heirloom Garden next time you're at the museum.