String literals like "Hello World!" are null-terminated, but char arrays are not automatically null terminated.

The general principle I've always taken is to be extra cautious and assign '\0' to the the end of the string unless that causes a performance problem. In those cases, I'm extra careful about which library functions I use.

In c, a string is not an intrinsic type. A C-string is the convention to have a one-dimensional array of characters which is terminated by a null-character, by a '\0'.

Using that knowledge it should be

 struct date{  char day[3];  char month[3];  char year[5];  

Otherwise scanf¹ won't be able to store the \0 within the array and it would be undefined behavior if you try to get 2 digit input (for day or month or 4 digit year by using %s format specifier) with it - because scanf will try to write it beyond the array and that would be Undefined Behaviour.

The scanf usage would be

if(scanf("%2s",current_date.day)!=1){
    fprintf(stderr,"Error in input\n");
    exit(EXIT_FAILURE);
}

The way you defined your structure - would give you 1 digit month, 1 digit day and 3 digit year if you store the corresponding \0 also. That is not what you want. In C strings are realized using nul terminated char array - scanf stores them. That's why in my case I have used %2s - so that the remaining space is there for storing \0.

¹ Here note one thing from reference under %s format specifier

matches a sequence of non-whitespace characters (a string) If width specifier is used, matches up to width or until the first whitespace character, whichever appears first. Always stores a null character in addition to the characters matched (so the argument array must have room for at least width+1 characters).

Note: As pointed by Simon Berthiaume in comment -

You can write string length like this: char year[4+1];, that way it is clear what the content size is intended to be. For example in this case it is 4 digit year that you wanted.

However, the fscanf at the end of the code only matches up to but not including the first null character.

That is incorrect, as is demonstrated by the fact that the output of the following program is “"Hello", then "world".” fscanf reads the entire line up to the new-line character; it does not stop at the null character.

#include <string.h>
#include <stdio.h>
#include <ctype.h>

int main(void)
{
    char inBuf[40] = {0};
    char outBuf[] = "Hello\0world\n";
    FILE *fp = fopen("MyFile.txt", "w+b");
    fwrite(outBuf, 1, sizeof(outBuf), fp);
    fflush(fp);
    rewind(fp);
    fscanf(fp, "%s", inBuf);
    printf("\"%s\", then \"%s\".\n", inBuf, inBuf + strlen(inBuf) + 1);
}

You can use c[i]= '\0' or simply c[i] = (char) 0.

The null/empty char is simply a value of zero, but can also be represented as a character with an escaped zero.