"shopt -s inherit_errexit" is important

a.bash

#!/usr/bin/env bash

set -o errexit
set -o nounset

f() {
  echo -n hello
  INVALID_COMMAND
  echo -n world
}

bar=$(f)
echo "bar is $bar"

# It will print
# ./a.bash:行8: INVALID_COMMAND：未找到命令
# bar is helloworld

b.bash

#!/usr/bin/env bash

set -o errexit
set -o nounset
shopt -s inherit_errexit

f() {
  echo -n hello
  INVALID_COMMAND
  echo -n world
}

bar=$(f)
echo "bar is $bar"

# It will print
# ./b.bash:行9: INVALID_COMMAND：未找到命令

@badrelmers Fixed. Thank you