What can a marine do to marry an undocumented immigrant?

I cannot comment on your boyfriend's career; he may want to consult a JAG attorney on this one before you get married. As to your immigration status, as the law stands right now, the problem is that you entered EWI (entered without inspection), regardless of your age and circumstances. This means that you cannot adjust status without leaving the US. If your boyfriend's career is unaffected and you two do get married, you will have to apply for and receive an unlawful presence waiver. The good news is that now you can wait for the waiver here in the US. Once it is granted (and in your case, it should be), you will have to travel to Mexico and go through the consular processing in order to return to the US as a wife of a US citizen. Currently, there is no other way. I cannot guarantee you that this process is a sure thing or how long you would have to be in Mexico, but I would plan on three months or so. Unless the anticipated immigration reform changes anything, this is the unfortunate long way for this process in your case - all because you were not admitted and inspected at the border. If you had a visa and you came in lawfully but then overstayed your welcome, you could adjust status after marriage without leaving the US. But all EWI cases must exit the US, unfortunately, in order to consular process. That stated, I would not be scared of the process: if you have no criminal history (no arrests or any other "hiccups" with the law), you should be able to get the waiver and from there, I see no problems with the consular processing. The only variable is the timing, as I cannot tell you how long you may have to be in Mexico. And I cannot answer your question as to whether your boyfriend's military career could suffer if he married an undocumented alien, so that is something for him to research through military lawyers.

You are not eligible for adjustment of status on the basis of marriage if you entered the US unlawfully. You may be eligible for DACA but the requirements for that and permanent residence are quite different.

Right now you cannot adjust status here in the US. Maybe the new immigration law will help if it is passed. But you can marry and your husband can file the I-130 for you. But you cannot file the I-485 because you were not inspected.