How did people at the time feel about Queen Elizabeth's ruling without a king?
I am not sure that the above answer is completely accurate. There was, of course, a woman ruler at the time of Elizabeth's reign, but this was Mary Queen of Scots who was not trusted in England or Scotland; in fact Mary was forced to flee Scotland ahead of a charge of involvement in the murder of her married lover's wife. While in England, Elizabeth kept Mary under house arrest until Mary was involved in a plot to overthrow Elizabeth, and was executed for it.
Furthermore, England had itself been ruled for five years by a previous Queen, Mary I, who was Elizabeth's half sister. So the idea of a female ruler was not entirely new to the people of England. Since Mary had no male heirs, it was commonly accepted and expected that Elizabeth would succeed her.
Additionally, Elizabeth proved to be exceptionally skillful at earning the respect and trust of the English people. While she accepted Parliament's declaration that she was
supreme governor of this realm in all things ecclesiastical and temporal
she would not accept the title of head of the church as her father, Henry VIII had done. She rather tactfully stated that this right belonged to God alone. She made herself visible to the common people quite often, and wore stunning attire when she did so. The one exception was when the Spanish Armada threatened, at which point she appeared before her troops dressed in full battle armor. The quote in the above post was actually made when she addressed her troops in preparation for battle. Elizabeth also was a gifted speaker and made a number of speeches to Parliament which were considered brilliant.
She wisely avoided religious conflict by adopting a hybrid religious doctrine that allowed Catholics and Anglicans to worship with some degree of comfiture. Her diplomacy and skill allowed her to rule for forty five years. As she grew older, she once commented to her people
Though God has raised me high, yet this I count the glory of my crown, that I have reigned with Your love….Your prosperity has been my chief concern….My heart was never set on worldly goods, but only for my subjects good.
Her subjects took her at her word. It is no small wonder that she was often referred to by the English people as "Gloriana."
People in England at the time of Queen Elizabeth's reign were somewhat worried about the idea of a female ruler, but not terribly so. There had been other examples of women ruling in England and in Scotland and so the idea was not completely foreign.
However, Elizabeth did have to be aware of her sex and the effect that it had on people. She had to, for example, reassure her subjects at one point that she really had the ability to rule. She once gave a speech in which she said
I know I have the body but of a weak and feeble woman; but I have the heart and stomach of a king, and of a king of England too...
From this, we can see that Elizabeth was aware of the need to reassure her subjects that she was as competent as a man. However, she did not face anything like rebellions because of her sex.