"Cannot stand" can mean "cannot resist" or "cannot hold the line", as in "stand up to" or "stand with".
The ballad is about what the Alliance did to the Browncoats.
Take my love
Take my land
"Take my love" refers to the people who where killed. "Take my land" refers to the Alliance conquering the Independents (and maybe specific acts of appropriation).
Take me where I cannot stand
Here, we use "Take me" in its double meaning -- to bring someone somewhere. "Bring me to the place where I cannot hold against your attack" -- Serenity Valley.
As @ToddWilcox has pointed out, you also "cannot stand" in space (barring artifical gravity). This reading generates parallels with the next stanza ("to the black"); the "no place" where you "can be" is outer space.
I don't care
I'm still free
You can't take the sky from me
The anthem of defiance. In a non-space faring culture, "the sky" is something that noone can occupy. In this case, it also refers to the freedom of flying around and not being subject to alliance rules all the time.
Take me out
To the black tell 'em I ain't coming back
I will flee to outer space.
Burn the land
And boil the sea
Refers both to bombardment, and takes the previous verse where they "take" it, and stresses they can even destroy it (goes a step further).
you can't take the sky from me
Somewhat similar meaning. When juxtaposed against the burn/boil lines, it says "you cannot destroy the entire sky, there is nothing for you to bombard".
Have no place
I can be
since I found Serenity
Serenity refers to a state of grace of acceptance. It also refers to Serenity valley, where the Browncoats where defeated; since then, they have no place they can remain independent of the Alliance. The third meaning is the ship Serenity (in the context of the TV show), where Mal feels like he has to wander because there is no place he feels he can stay. The forth interpretation is "no place" is outer space; the singer can be in outer space ("no place").
but you can't take the sky from me
Previous meanings, but the Firefly Ship Serenity makes the "I can stay in space and fly around and I can be free" attach to "found Serenity" stronger.
In the world fiction, that part (that the ship is part of the Lyric's meanings) probably doesn't apply to the authorial intent (unless one of the crew wrote the song); it still could have that meaning to Mal, who took the other layers of meaning of that ballad and was part of the inspiration to name the ship (reversing the cause).
As an alternative explanation, the first stanza could be what led the Browncoats to war; the Alliance started enforcing their law: killing Independents, claiming land that the Independents claimed or disagreeing with who owned what, until they reach the point where the Independents "could not stand" to suffer the Alliance's oppression.
They are still free. They cannot take the sky -- the Independents rose up, and fought back from planet to planet.
The second stanza would then refer to the war itself and its aftermath, where the Alliance bombarded Independent colonies (burn the land and boil the sea). The Independents had to retreat into deeper space "Take me out to the black tell'em I ain't coming back." and could not hold. They fled their planets (you can't take the sky from me), ever since the battle of Serenity.
The eventual cleanup would then occur, but not be covered by the rebel verse.
Like most pieces of poetry, it admits multiple interpretations.
Here is a footnote in a book to another book (which I don't have a URL to) where Joss states that it was penned "so that it could be sung as a Civil War lament". The source (in "The Philosophy of Joss Whedon) is claimed to be: (Firefly, the Official Companion, vol 2. (London: Titan 2007), 33)