I guess you want to make arrays of pointers instead of arrays of characters.
Try this:
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include <string.h>
int main(void){
int randomNum;
const char *takenWords[4];
const char *words[20]={"DOG", "CAT", "ELEPHANT", "CROCODILE", "HIPPOPOTAMUS", "TORTOISE", "TIGER", "FISH", "SEAGULL", "SEAL", "MONKEY", "KANGAROO", "ZEBRA", "GIRAFFE", "RABBIT", "HORSE", "PENGUIN", "BEAR", "SQUIRREL", "HAMSTER"};
srand(time(NULL));
for(int i=0; i<4; i++){
int dupe=0;
do{
randomNum = (rand()%20);
takenWords[i]=words[randomNum];
dupe=0;
for(int j=0;j<i;j++){
if(strcmp(words[randomNum],takenWords[j])==0)dupe=1;
}
}while(dupe);
printf("%s\n", words[randomNum]);
}
getchar();
return 0;
}