Well, I have found enough time to take college classes while I was on Active Duty (with 2 wars were going on), but most of them came while I was on Shore Duty (on Sea Duty, I took 1 class per semester. For Shore Duty, I physically attended classes on a full time basis; actually, I did it more than most full time students: Spring Quarter, Summer Quarter, Autumn Quarter, Winter Quarter, & Winter Inter-session).
Once you get to your first command (after boot camp & pipeline), you will still need to get the EWS Qualification (to be a Senior EOD) before you are allowed to take any college classes (including boot camp & pipeline, that would be the first 3-4 years of your first enlistment).
Regardless of the job, you will have to complete whatever qualifications that the Navy requires you to do before you are allowed to take none-Navy classes.
Taking classes while you are on Active Duty & Sea Duty is very tough; but if you budget your time wisely and be disciplined about sticking to the schedule that you set (except when duty calls), then it is possible to accomplish your goal. Keep in mind that you will also have to sacrifice some of your social life & sleep as well (but you will still need to have a bit of a social life and sleep to keep your sanity).
Former Navy EOD Tech. & struggled to get B.S. in Accounting & Finance (and MBA after the Navy).