Unexpected EOF and syntax error

The problem is, that your case statement is missing the subject - the variable which it should evaluate. Hence you probably want something like this:

#!/bin/bash
cat <<EOD
choose one of the following options:
1) display all current users
2) list all files
3) show calendar
4) exit script
EOD

while true; do
    printf "your choice: "
    read
    case $REPLY in
        1) who;;
        2) ls -a;;
        3) cal;;
        4) exit;;
    esac    
done

Here case uses the default variable $REPLY which read fills when it's not given any variable names (see help read for details).

Also note the changes: printf is used to display the prompt in each round (and doesn't append a newline), cat is used to print instructions on several lines so that they don't wrap and are easier to read.

Let's not forget select:

choices=( 
    "display all current users" 
    "list all files" 
    "show calendar" 
    "exit script"
)
PS3="your choice: "
select choice in "${choices[@]}"; do
    case $choice in
        "${choices[0]}") who;;
        "${choices[1]}") ls -a;;
        "${choices[2]}") cal;;
        "${choices[3]}") break;;
    esac
done

Let us try for a single case initially. I will use read -p to read the user input into a variable opt followed by case statement as below.

#!/bin/bash
read -p "choose one of the following options : \
  1) display all current users \
  2) list all files \
  3) show calendar \
  4) exit script" opt
case $opt in
1) who;;
2) ls -a;;
3) cal;;
4) exit;;
esac

The above script works fine and now, I believe you need to have it in a loop so that you can read user input until the user presses option 4.

So, we could do it with a while loop as below. I set the variable opt with the initial value as 0. Now, I am iterating in the while loop as long as the opt variable has a value as 0 (which is why I reset the opt variable as 0 at the end of the case statement).

#!/bin/bash
opt=0;
while [ "$opt" == 0 ]
do
read -p "choose one of the following options : \
  1) display all current users \
  2) list all files \
  3) show calendar \
  4) exit script" opt

case $opt in
1) who;;
2) ls -a;;
3) cal;;
4) exit;;
esac
opt=0
done

I would personally put the "while" at the beginning of the code. If you then follow it with a :, it will allow it to loop around as many times as you wish. This is how I would write it.

while :
do
    echo "choose one of the following options : \
      1) display all current users \
      2) list all files \
      3) show calendar \
      4) exit script"
    read string
    case $string in
        1)
            who
            ;;

then carry on the questions, and end with

esac
done