"Romeo and Juliet" is generally believed to take place in the fourteenth or fifteenth century. Most of the literature books that I have used and taught from have given the time period as the 1300's (fourteenth century). As an English teacher, I have been teaching the play for nine years now and always give the setting as the 1300s.
Although Shakespeare wrote Romeo and Juliet during the 1590s, and the work was eventually published in approximately 1597 or 1599, the play itself is set much earlier, most likely in the 1300s.
It is important to note that Shakespeare was not the first writer to pen the story of Romeo and Juliet. Works such as Arthur Brooke's poem "The Tragicall Historye of Romeus and Juliet" (1562), William Painter's The Palace of Pleasure (prior to 1580), Matteo Bandello's Giulietta e Romeo (1554), Masuccio Salernitano's Il Novellio, and/or Luigi Da Porto's A Story Newly Found of Two Noble Lovers. Knowing this helps us understand that the dramatization was not meant to be set within Shakespeare's own day and age; rather, it was most likely set during the lifetime of the real-life lovers who inspired the narrative, who lived in Verona in 1303.
It is believed that Shakespeare wrote Romeo and Juliet in the 1590s (most likely 1594). He wrote it during the reign of Queen Elizabeth I. The play was probably published in 1597 and/or 1599. In fact, it was one of the first plays performed when theaters reopened after being closed due to the plague. It is assumed that Shakespeare took several years to write it but there isn't a consensus on this information. We aren't exactly sure about these dates. These are calculations and assumptions made by historians. It is known that Shakespeare wrote many of his plays between 1594 and 1603.
The story itself takes place in either the 14th (1300-1400) or 15th century (1401-1500). We know the month/season specifically. It took place in the summer, in the month of July. (In Scene III, it is mentioned that it is a fortnight and odd days away from Juliet's birthday which falls on "Lammas-eve" and that is believed to be in August or late July. A fortnight is two weeks.)
On a side note, it is thought that Romeo and Juliet is based on the real life story of two lovers who lived in Verona, Italy around 1303.
It's generally thought to be set in the 14th century, so the 1300s. Hope this helps.
It takes place during the black plague