In order to put the strings in Program Space, you have to have explicit declarations for each string, and put each string in Program Space:

char string_1[] PROGMEM = "String 1";
char string_2[] PROGMEM = "String 2";
char string_3[] PROGMEM = "String 3";
char string_4[] PROGMEM = "String 4";
char string_5[] PROGMEM = "String 5";

Then use the new symbols in your table, like so:

PGM_P string_table[] PROGMEM =
{
string_1,
string_2,
string_3,
string_4,
string_5
};