We read Early Modern English such as Shakespeare in its original form, and even though it's a bit harrowing we can generally make it out. We can also go back to Middle English such as Chaucer in its original form, and though it is more difficult, we can generally make it out. With Old English works such as Beowulf, one would require a translation as it's practically unintelligible.
Would someone in Chaucer's time or Shakespeare's time have been able to look at a piece of Old English, and be able to read it just as easily as we read Chaucer/Shakespeare?
I think it would depend on the person trying to read it, just like it does today with a person reading Shakespeare or Chaucer. The more one has read and studied, the easier it would have been. But the average person would have had a lot of difficulty.
Already by the thirteenth century, Old English was getting difficult for the average person to read, but the scribe who wrote in the so-called "Tremulous Hand" of Worcester was able to work through Old English manuscripts and provide glosses to help.
If we move forward to the 16th and 17th centuries, there are scholars who studied Old English. People like Humphrey Wanley (who catalogued the collection of manuscripts once owned by Robert Cotton), John Leland and other antiquaries could read it as well, after studying it.
Even if one could understand the words and could handle the grammar (which is much more different than Middle English is for us today), the process of actually reading it would have been quite difficult, simply because of the strange letter forms. I have seen some laws from the Middle English period in which an earlier law of William the Conqueror was quoted and it is clear that the scribe was unable to make much sense of the text he was copying. When it came to odd letters like the g ð and þ he simply tried to replicate the form of the letter, not realizing what it meant. Besides the letter forms, the spelling conventions were a lot different. We spell things today roughly similar to how Middle English was spelled, but Old English has a very different way to spell sounds than we do. Besides the odd letters like ð and þ, in Old English "sc" represents the "sh" sound, "cg" is our "j" sound, and so on.
TL:DR It would be much more difficult for a Middle English speaker to read Old English than a Modern English speaker to read Middle English.
You may also want to ask the linguists in /r/AskLinguistics, /r/Linguistics, or /r/AskSocialScience.
I asked a very similar question on r/linguistics a couple of months ago and received some good responses
No; hardly anyone could read at that time. You'd have to be a priest, and educated to read Old English in particular. Middle English was different enough that people would not readily understand Old English.