In Shakespeare's classic Romeo and Juliet, first performed in 1597, Romeo meets Juliet at the Capulet ball on Sunday evening. How do we know that this event occurred on Sunday? On the following day, when Lord Capulet arranges for Paris to marry Juliet, he asks Paris what day it is (3.4.20). Paris responds, "Monday, my lord" (3.4.21).

Earlier on Monday, Romeo went to Friar Lawrence and arranged his marriage to Juliet. He then asks the Nurse to tell Juliet to come to Friar Lawrence's abbey that afternoon: "Bid her devise / Some means to come to shrift this afternoon. / And there she shall at Friar Lawrence' cell / Be shrived and married" (2.4.84-87). Therefore, Romeo and Juliet marry within twenty-four hours of knowing each other.

How long did they know each other before they both died, however? Lord Capulet initially establishes that Juliet and Paris's wedding will take place on Thursday. He says, "O’ Thursday let it be.—O’ Thursday, tell her, / She shall be married to this noble earl" (3.4.23-24). "This noble earl" refers to Paris, since he is the cousin of Prince Escalus. 

This is where things get a little confusing. Since Juliet and Paris's wedding is planned for Thursday, Friar Lawrence advises Juliet to drink the poison on Wednesday night:

"Hold, then. Go home; be merry; give consent
To marry Paris. Wednesday is tomorrow.
Tomorrow night look that thou lie alone;
Let not the Nurse lie with thee in thy chamber.
Take thou this vial, being then in bed,
And this distilling liquor drink thou off" (4.1.91-96).

This scene must occur on Tuesday, since Friar Lawrence states that Wednesday is the following day.

Later on Tuesday, however, Lord Capulet changes the date of the wedding to Wednesday. When Juliet tells Lord Capulet that she now wishes to marry Paris, Lord Capulet says, "Send for the county. Go tell him of this. / I’ll have this knot knit up tomorrow morning" (4.2.21-22). With this, Lord Capulet moves the wedding to Wednesday, presumably to prevent Juliet from resisting Paris again. Later that night, as Juliet prepares to drink the poison, she says, "What if this mixture do not work at all? / Shall I be married then tomorrow morning?" (4.3.21-22). This event thereby occurs on Tuesday night.

On Wednesday morning, the Capulets discover Juliet's body and place her in the crypt. On Friday morning, Romeo enters the crypt, mistakenly presumes that Juliet is actually dead, and kills himself. Juliet then wakes up and does the same. Two watchmen enter, and the first says, "Pitiful sight! here lies the county slain, And Juliet bleeding, warm, and newly dead, Who here hath lain these two days buried" (5.3.188-190). Since Juliet was placed in the crypt on Wednesday morning, the time of Juliet's death must be very early on Friday morning. Romeo and Juliet therefore knew each other for a little more than four days before they met their tragic ends.

For more information, please check out the eNotes guide to this great play linked below!

The short answer is that they'd known each other for less than 24 hours when they were married.  They meet at the party late at night and are married the following afternoon by Friar Laurence.

The long answer is that Romeo is over the moon for Rosaline at the outset of the play but once he goes to the party at the Capulets and see Juliet, he forgets her completely.  Once they have the chance to speak to each other, both of them pledge undying love for each other and lament that they cannot immediately be together.

Once Romeo leaves the party, he immediately goes to see Friar Laurence to see if he can convince him to marry the two.  It is only a few short hours later that he gets Juliet to come, summoned through her maid, and Friar Laurence marries them.