Get free disk space with df to just display free space in kb?

To get the output of df to display the data in kb you just need to use the -k flag:

df -k

Also, if you specify a filesystem to df, you will get the values for that specific, instead of all of them:

df -k /example

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Regarding the body of your question: you want to extract the amount of free disk space on a given filesystem. This will require some processing.

Given a normal df -k output:

$ df -k /tmp
Filesystem     1K-blocks    Used Available Use% Mounted on
/dev/sda1        7223800 4270396   2586456  63% /

You can get the Available (4th column) for example with awk or cut (previously piping to tr to squeeze-repeats (-s) for spaces):

$ df -k /tmp | tail -1 | awk '{print $4}'
2586456
$ df -k /tmp | tail -1 | tr -s ' ' | cut -d' ' -f4
2586456

As always, if you want to store the result in a variable, use the var=$(command) syntax like this:

$ myUsed=$(df -k /tmp | tail -1 | awk '{print $4}')
$ echo "$myUsed"
2586456

Also, from the comment by Tim Bunce you can handle long filesystem names using --direct to get a - instead, so that it does not print a line that breaks the engine:

$ df -k --direct /tmp
Filesystem     1K-blocks    Used Available Use% Mounted on
-                7223800 4270396   2586456  63% /

You can use stat(2) command to display free blocks and also to find out how large each block is, e.g.

stat -f --printf="%a %s\n" /

will display number of free blocks (%a) on a given file system (/) followed by a block size (%s). To get size in kB, you can use bc(1) command as in the following example:

stat -f --printf="%a * %s / 1024\n" / | bc

Finally, to put this into a variable is just a matter of using backtick substitution (or $() as in the first answer):

SOMEVAR=`stat -f --printf="%a * %s / 1024\n" / | bc`