Background: Some think that in islam unmarried men and women are completely restricted and cannot be alone together under any circumstances.

The Quran gives an example in which an engaged couple may meet in secret if they have something righteous to discuss:

And there is no sin upon you if you openly propose marriage to these women, or you keep it between yourselves. God knows that you will be thinking of them, but do not promise them secretly, except that you say a saying honourable/befitting... [2:235]

The Quran also cites an example in which if an unrelated male believer is alone inside the prophet's house and needed to ask his wives for something, they should do so from behind a barrier:

...If you ask his wives for something, ask them from behind a barrier*... [33:53]
*Note, the male is requested to ask from behind a "barrier/hijab", not the female. The term "hijab" denotes anything that intervenes between two things or prevents something, e.g. screen, partition, curtain, obstacle.

It is later clarified in 33:55 that 33:53 is about unrelated male believers only:

There is no sin upon them before their fathers, or their sons, or their brothers, or the sons of their brothers, or the sons of their sisters, or their women, or those who are committed to them by oath (e.g. servants or dependents in the household). And be aware of God, for God is witness over all things. [33:55]

Similarly, Abraham and his wife were in the presence of male guests in their home, freely participating in conversation, laughing [11:69-76].

There is also a clear example in The Quran of Moses speaking to two unmarried women in public, one of whom he married later:

And when he came to the water of Madyan, he found on it a group of men watering, and he found besides them two women keeping back (their flocks). He said: What is the matter with you? They said: We cannot water until the shepherds take away (their sheep) from the water, and our father is a very old man. So he watered (their sheep) for them, then went back to the shade and said: My Sustainer! surely I stand in need of whatever good Thou may send down to me. Then one of the two women came to him walking bashfully. She said: My father invites you that he may give you the reward of your having watered for us... [28:23-25]

And there is the example of Solomon and the Queen of Saba:

She was asked to enter the lofty Palace: but when she saw it, she thought it was a lake of water, and she uncovered her shins. He said: "This is but a palace paved smooth with slabs of glass." She said: "O my Lord! I have indeed wronged my soul: I submit, with Solomon, to the Lord of the Worlds." [27:44]

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