How do I take a variable sized string as input using only getchar without asking user for string size before?

You'll want to use realloc to extend the input buffer, although you won't want to do it for every individual character (it's a relatively expensive operation, and can result in the string being moved around in memory). A common trick is to double the size of the buffer as you reach the end of it, so as you read characters the buffer size goes from 16 to 32 to 64, etc., minimizing the number of realloc calls. The tradeoff is a little internal fragmentation - you may wind up storing 65 characters in a 128-character buffer. But on average, that shouldn't be too much of a problem. Here's an example:

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

#define START_SIZE 16 // some size that should handle most cases

/**
 * Get the next line from the specified input stream.  Return characters up to
 * (but not including) the next newline character or EOF.  Return the size of the
 * allocated buffer as well.  
 */
char *getline( FILE *stream, size_t *size )
{
  *size = START_SIZE;
  size_t i = 0;

  /**
   * Initial allocation, buf can store a string up to START_SIZE - 1 characters.
   * If initial allocation fails, return NULL.
   */
  char *buf = malloc( sizeof *buf * *size );
  if ( !buf )
  {
    fprintf( stderr, "Failure to allocate initial buffer\n" );
    return NULL;
  }

  /**
   * Read from the input stream until we see a newline or EOF.  Newline will
   * *not* be stored in the returned string.
   */
  for ( int c = fgetc( stream ); c != '\n' && c != EOF; c = fgetc( stream ))
  {
    /**
     * Have we hit the end of the input buffer yet (allowing for the terminator)?
     */
    if ( i + 1 == *size )
    {
      /**
       * Yes.  Double the size of the buffer using realloc.  
       * If realloc cannot satisfy the request, it will return 
       * NULL and leave the contents of buf unchanged.  Therefore,
       * we want to make sure we assign the result to
       * a temporary variable and check it, otherwise we
       * could potentially lose our reference to the
       * previously allocated memory, leading to a memory leak.
       */
      char *tmp = realloc( buf, sizeof *buf * (*size * 2));
      if ( tmp )
      {
        buf = tmp;
        *size *= 2;
      }
      else
      {
        fprintf( stderr, "Unable to extend buf, returning what we have so far\n");
        return buf;
      }
    }
    buf[i++] = c;
    buf[i] = 0; // zero terminate the string as we go
  }
  return buf;
}

int main( void )
{
  size_t bufsize;
  printf( "Gimme a string: ");
  char *str = getline( stdin, &bufsize );
  printf( "You entered: \"%s\"\n", str );
  printf( "length = %zu, buffer size = %zu\n", strlen( str ), bufsize);
  free( str );
  return 0;
}

And some sample runs:

john@marvin:~/Development/getline$ gcc -o getline -std=c11 -pedantic -Wall -Werror getline.c

john@marvin:~/Development/getline$ ./getline
Gimme a string: this
You entered: "this"
length = 4, buffer size = 16

john@marvin:~/Development/getline$ ./getline
Gimme a string: this is a test
You entered: "this is a test"
length = 14, buffer size = 16

john@marvin:~/Development/getline$ ./getline
Gimme a string: this is a test of
You entered: "this is a test of"
length = 17, buffer size = 32

john@marvin:~/Development/getline$ ./getline
Gimme a string: this is a test of the emergency broadcast system.
You entered: "this is a test of the emergency broadcast system."
length = 49, buffer size = 64

john@marvin:~/Development/getline$ ./getline
Gimme a string: this is a test of the emergency broadcast system.  in the event of an actual emergency, you would be dead by now.  
You entered: "this is a test of the emergency broadcast system.  in the event of an actual emergency, you would be dead by now.  "
length = 115, buffer size = 128