Well, there's a few different things to address in answering this.

• Shakespeare wasn't writing in the Middle Ages. He was writing in the Early Modern period.

It sounds a bit like nitpicking, but it's an important distinction because the era was one in which language changed a lot. Read some Shakespeare, and read something from a medieval source like Chaucer and you'll see the difference in how much effort it takes a modern English speaker.

For instance, let's take a bit of the prologue from the Canterbury Tales (late 14th century):

Ther was also a Nonne, a PRIORESSE, That of hir smyling was ful simple and coy; Hir gretteste ooth was but by seynt Loy;
And she was cleped madame Eglentyne. Ful wel she song the service divyne, Entuned in hir nose ful semely; And Frensh she spak ful faire and fetisly, After the scole of Stratford atte Bowe,
For Frensh of Paris was to hir unknowe.

So there are a couple of interesting things there. "She was clept" (ie called) is a hangover from Anglo-Saxon. Shakespeare does actually use it occasionally but it's reasonably rare so he's clearly making a stylistic choice when he does so.

Also notice that the Prioresse speaks French "fair and fetisly" (skilfully) but it's the sort that she learned in England at Stratford-at-Bowe and "Frensh of Paris" is unknown to her. So there is still a particular Anglo-French dialect which is spoken by educated folk and completely different from the French spoken in France.

Now let's jump forward 200 years and look at the late 16th century prologue to Henry V.

O for a Muse of fire, that would ascend The brightest heaven of invention, A kingdom for a stage, princes to act And monarchs to behold the swelling scene!
Then should the warlike Harry, like himself, Assume the port of Mars; and at his heels, Leash'd in like hounds, should famine, sword and fire Crouch for employment. But pardon, and gentles all, The flat unraised spirits that have dared On this unworthy scaffold to bring forth So great an object: can this cockpit hold The vasty fields of France? or may we cram Within this wooden O the very casques That did affright the air at Agincourt?

So hopefully that reads a little easier. There are still some words with changed or unfamiliar meanings ("assume the port of Mars" - ie "look all military and stuff") but it reads much closer to modern English.

Anyway, that's all just a bit of an aside.

• So I guess the meat of the question is: did ordinary folk at the time speak the way Shakespeare writes?

And the answer is "not really". In the same way as people in real life don't talk like a Tarantino movie. Dialogue is simplified; people don't um and ah, talk over each other, lose their thread, fail to express themselves. Tarantino is famous for his dialogue, Woody Allen is another who is praised for realistic sounding dialogue but again it's always going to be a heightened version of reality. You go see Tarantino, or Whedon, for the snappy dialogue - to enjoy the snarky comments of a Tony Stark or the stylish bible-quoting of Jules Winnfield. But you don't expect people to actually talk like that.

In fact, in Shakespeare's day, one of the big draws of the theatre was the great speeches, the verbal sparring and the creativity in the use of language, simile and metaphor. Look at Romeo and Juliet - in the early stages of the play, the banter between Romeo, Mercutio and the others is really extraordinary; they're taking what the other says, twisting the meaning and throwing it back. Rap isn't really my medium, but you can hear Lin-Manuel Miranda doing the same thing in Hamilton.

That said of course, he was writing in a heightened version of everyday language - we appreciate and applaud the verbal dexterity of a rapper, but it's not like they're using language that people don't understand, for the most part.

You also have to consider the differing education levels of varying parts of the audience - some would be very highly-educated in the classics, and would appreciate the references to ancient myths and legends, others wouldn't get those at all but would enjoy the bawdier bits and the heroic speeches and so on.

So the question of "did contemporaries really talk like that" is a difficult one to answer simply; very few would have spoken exactly like the dialogue in a Shakespeare play, but most would have recognised their own way of speaking in there. And yes, it seems "fancy" to us, just as people in 50 or 100 years time will watch Pulp Fiction and say "God, they were so stuffy and stilted back then. Did they really talk like that?". Watch "Brief Encounter" is

Sources:

• The "Arden" editions of Shakespeare plays are great for information about how a specific play was received, details of performances and so on.

• Shakespeare's Words by Ben and David Crystal is a great survey of Shakespeare's language