You need 11 men on offense and 11 men on defense so ideally, without backups, 22 minimum and even that assumes one of them can kick. No modern school would go to war with just 13 players. A 13 man team means everyone played both sides of the ball and playing a whole season means nobody got seriously injured.
My high school team had about 60 kids, but most of the playing was done by the same 11 guys.