I will not answer for the cadet question, but I know as an Officer you can. Obviously it will depend on the country, but overall the answer is yes. We knew many officers stationed overseas that could not get base housing and opted to buy.
You will need to inform the command and get their approval. Also it will be investigated for your security clearance.
This is also true for businesses. The main reason why is due mainly to the fact of image. Image not only because you are expected as an officer to conduct yourself in a certain manner, but also image in the fact that the business cannot be tied to them, looking like they play favorites to members and have connections with them...Think non-compete.
I don't know what business you are thinking about, but you are very young. Anyone with age and experience will tell you even running a small business takes time and dedication. A LOT OF IT, if you want to earn a profit.
Your job right now is education and serving in the military. Many HS srs believe that because school has always been easy and had tons of time to play XBox or hang with friends that this will continue. This is a fallacy. As a cadet and a newly minted officer, you will be hitting the books often and for long periods.
Want to fly or go SEALS/DELTA. People will tell you it is hard, hard, hard for the 1st yr or two. Downtime is Friday night and by Sunday morning you are back to the grindstone studying. Trying to juggle a business and the military may result in failure for one or both.
Additionally, this business would be overseas. While you are at school, even at WP/USNA and the property is in the UK, they are finishing their business day. While you are studying they are sleeping. That leaves you getting up early to email/phone before their day begins...think 3 a.m. AFA think staying up to midnight.
Personally, I cannot think of any business where this is a doable scenario because we haven't even discussed deployments to places like Afghanistan. Even if your business was to raise sheep and sheer them for wool with an overseer, it would still take time out of your day to discuss bills, revenues and status. We had friends with successful businesses that actually took a pay cut to join the military, within months they sold the business because they couldn't juggle both. Small business owners, even if it is an internet business spend 7 days a week 8-12 hours a day grooming it to be successful.