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On Linux and with shells that implement here-documents with writable temporary files (like bash does), you can do:

{
  out=$(
    chmod u+w /dev/fd/3 && # needed for bash5+
      ls /dev/null /x 2> /dev/fd/3
  )
  status=$?
  err=$(cat<&3)
} 3<<EOF
EOF

printf '%s=<%s>\n' out "$out" err "$err" status "$status"

On Linux and with shells that implement here-documents with writable temporary files (like zsh or bash versions prior to 5.1 do), you can do:

{
  out=$(
    chmod u+w /dev/fd/3 && # needed for bash5.0
      ls /dev/null /x 2> /dev/fd/3
  )
  status=$?
  err=$(cat<&3)
} 3<<EOF
EOF

printf '%s=<%s>\n' out "$out" err "$err" status "$status"

{
  IFS= read -rd '' err
  IFS= read -rd '' out
  IFS= read -rd '' status
} < <({ out=$(ls /dev/null /x); } 2>&1; printf '\0%s' "$out" "$status")

{
  IFS= read -rd '' err
  IFS= read -rd '' out
  IFS= read -rd '' status
} < <({ out=$(ls /dev/null /x); } 2>&1; printf '\0%s' "$out" "$?")

{
  out=$(ls /dev/null /x 2> /dev/fd/3)
  status=$?
  err=$(cat<&3)
} 3<<EOF
EOF

printf '%s=<%s>\n' out "$out" err "$err" status "$status"

{
  out=$(
    chmod u+w /dev/fd/3 && # needed for bash5+
      ls /dev/null /x 2> /dev/fd/3
  )
  status=$?
  err=$(cat<&3)
} 3<<EOF
EOF

printf '%s=<%s>\n' out "$out" err "$err" status "$status"