Being in the military often presents unique challenges for staying in touch, which is why Google has a special program set-up to giving away priority invites to military personnel. If you have a .mil email address just sign up here and Google will send you an invite within a day or two.
If you don’t have a .mil address, you can request an invite here. I submitted my request for an invite over six months and just got the reply a few days ago. If you don’t want to wait that long, you might consider paying for an invite–the going rate on eBay is $1. If you don’t want to spend a buck, then I suggest doing a “google voice” search on Twitter and finding someone who is willing to send you an invite for free. I did that last night for my roommate and it work! NOTE: You need to be in the US and have a US phone to sign up for Google Voice.