This sparked my curiosity so I went searching.
And from what I've read... you could certainly have a character in a story set in Europe or the Mediterranean picking wild strawberries even if your story was set as far back as pre-historic times.
Quoted from
http://www.vegparadise.com/highestperch45.html :
"Prehistoric man had little value for the wild strawberry because of its inconvenience, though he certainly did consume his share. The plants were found in the woods, often covered by lush overgrowth. The season was brief, only a few weeks, and the berries were smaller than our wild strawberries today. Hardly worth the trouble, since early man did not know how to preserve and store foods. Yet, a few tiny strawberry seeds were discovered by archeologists in Mesolithic sites in Denmark, Neolithic sites in Switzerland, and Iron Age sites in England."
Or if your story was set in France, you could have a strawberries being cultivated (on a small scale) in the 1300s.
Again quoted from
http://www.vegparadise.com/highestperch45.html :
'Sporadic efforts of strawberry cultivation began in the 1300's with a few plantings of the wild fruits into home gardens. On a grander scale, King Charles V adorned his Parisian gardens at the Louvre with 1200 strawberry plants in 1368. A few years later, the Duke and Duchess of Burgundy had their gardener plant a four-block area of their property near Dijon with strawberries."