I understand the pain you're feeling in discovering that this young woman is being what you perceive as perhaps cold and clinical in evaluating your relationship together. She's certainly trying to walk the line of believing that long distance relationships don't usually work. Some do, but the majority do not, especially when the distance is so vast and opportunities for getting together so few and far between.
Just like what you've heard from her, I know this isn't what you want to hear and I'm sorry. I would like to give better news to a serviceman
The issue of age is sort of a non-factor. 5 years difference is not significant enough to warrant a comment by anyone and I suspect this sister is just teasing her. If you were only 1 year older than she is I would wager the sister would make the same silly comment.
Ask your young lady if she'd at least stay in touch via mail, email, phone calls, etc.. You can still grow your relationship this way and stay in the forefront of her thoughts making it more feasible that when you both do have time to meet somewhere in this great big world that you'll know each other even better.
Don't force it - tell her you will respect her decision despite how much it hurts and that you hope staying in touch is something she'll compromise in doing.
Then just let it play out. If you stay in touch enough and let her pour her heart out in communicating with you - it will create a bond perhaps even strong than if you were physically together.
It's worth the effort and you have nothing to lose. It's really the only option.
And by the way, thank you for your service and God bless you on your tour