This depends on whether you want to store the command's output (either stdout, or stdout + stderr) or its exit status (0 to 255, with 0 typically meaning “success”).

To capture the output, you use command substitution:

output=$(command)      # stdout only; stderr remains uncaptured
output=$(command 2>&1) # both stdout and stderr will be captured

If you want both:

output=$(command)
status=$?

The assignment to output has no effect on command's exit status, which is still in $?.

If you want to capture stdout as well as taking action on success/failure, without explicitly storing or checking $?:

if output=$(command); then
    printf "it succeeded\n"
    ...

What you cannot do is capture stdout in one variable, and stderr in another, using only FD redirections.

You must use a temporary file (or a named pipe) to achieve that one.

Well, you can use a horrible hack like:

cmd() { curl -s -v http://www.google.fr; }
 
result=$(
    { stdout=$(cmd) ; } 2>&1
    printf "this line is the separator\n"
    printf "%s\n" "$stdout"
)
var_out=${result#*this line is the separator$'\n'}
var_err=${result%$'\n'this line is the separator*}

Obviously, this is not robust, because either the standard output or the standard error of the command could contain whatever separator string you employ.

And if you want the exit code of your cmd (here a modification in the case of if the cmd stdout nothing)

cmd() { curl -s -v http://www.google.fr; }
 
result=$(
    { stdout=$(cmd); returncode=$?; } 2>&1
    printf "this is the separator"
    printf "%s\n" "$stdout"
    exit "$returncode"
)
returncode=$?
 
var_out=${result#*this is the separator}
var_err=${result%this is the separator*}