In Chapters 5–6 of John Steinbeck's novel Of Mice and Men, what was Lennie and George's dream? Did they achieve it? Did the dream change as a result of obstacles or disillusionment?

George and Lennie’s dream was to have land of their own, but they were not able to achieve it because of their circumstances.
George keeps Lennie and himself motivated with dreams of having a ranch of their own one day.  It is like a bedtime story for the childlike Lennie: land of their own where they can settle down, and Lennie can tend the rabbits.  For George, it means an end to the migrant lifestyle.  For Lennie, it means no more judgement, and rabbits to pet.
Although Lennie’s understanding of the world is limited, he does realize that he is judged by others.  He knows, for example, that when he kills the puppy accidentally it will upset George.  In Lennie’s mind, it will interfere with the dream of land and the rabbits that go with it.   That is a cause and effect that Lennie understands.

"Why do you got to get killed? You ain't so little as mice." He picked up the pup and hurled it from him. He turned his back on it. He sat bent over his knees and he whispered, "Now I won't get to tend the rabbits. Now he won't let me." (Ch. 5) 

When Curley’s wife enters, he tells her that he is not going to talk to her.  Again, he won’t be able to tend the rabbits if he does.  She tells him she is lonely and George won't let him talk to her because he doesn’t want Curley to be mad.  George told Lennie to avoid Curley because he was the boss’s son.  He didn’t want them to lose their jobs, and was afraid that Curley would pick a fight because he was small and small men liked to pick fights with Lennie. 
Curley’s wife visits Lennie because she is lonely.  She tells him that she could have had a different life.  She could have been in movies and had nice clothes, and instead she is reduced to visiting Lennie in the barn because he is the only one who will even talk to her.  Lennie shares his dream with Curley’s wife, just as she shares hers with him. 

"We gonna have a little place," Lennie explained patiently. "We gonna have a house an' a garden and a place for alfalfa, an' that alfalfa is for the rabbits, an' I take a sack and get it all fulla alfalfa and then I take it to the rabbits." (Ch. 5) 

She tells him he is “nuts,” but unfortunately offers to let him stroke her hair since he is so infatuated with soft things.  She is desperate for attention.  He is lonely too.  In that moment, both of their dreams are quashed.  He strokes her hair—and breaks her neck. 
Thus the real obstacle to George and Lennie’s dream was that it was an impossible dream.  It was a fantasy with no basis in reality.  As much as George tried to protect Lennie from himself, it was not possible.  Lennie’s childlike innocence was a contrast to his strength.  He never realized what he was doing, and when George was not there to stop him, he was a danger to society. 
When Candy and George find the body, George’s reaction is resignation, and not resentment or sadness. 

George said softly, "-I think I knowed from the very first. I think I know'd we'd never do her. He usta like to hear about it so much I got to thinking maybe we would."
"Then- it's all off?" Candy asked sulkily.  (Ch. 5) 

George’s acknowledgement that the dream was never possible comes from a long line of disappointments.  He has spent his life taking care of Lennie.  Lennie was his bastion against loneliness, as well as a constant frustration.  He was also his responsibility.  George realized that he had to protect Lennie in the only way had left.
Even before he shot Lennie, George called upon the vision of their dream to calm him.

George had been listening to the distant sounds. For a moment he was businesslike. "Look acrost the river, Lennie, an' I'll tell you so you can almost see it." (Ch. 6)

Of course, George knew at this point that the dream was not to be.  It was a bittersweet illusion to carry Lennie into the afterlife.  George was putting Lennie out of his misery, like Candy allowing his old dog to be shot.  He didn't want to do it, but he had to.  It was better than the alternative.
For men who have nothing, sometimes a dream is all it takes to keep going.  The American Dream is to have land of one's own.  During the Great Depression, it was even more out of reach.  George and Lennie, as migrant farm workers, drew upon that dream for motivation.  It held them together.  It bonded them with others, such as Candy and Crooks.  Ultimately, it slipped through their fingers, because it was never real at all.