Here’s a nice & cheap sub for sale.
[QUOTE=mlees]
Is there any international laws prohibiting nuclear powered, privately owned vessels?
[/QUOTE]
Nope: NS Savannah - Wikipedia & Nuclear marine propulsion - Wikipedia.
Although the world is not really ready for this idea.
Can I suggest that you christen the submarine “The Tomnbarbara” or maybe “The Margoandjerry”?
I think it’s fitting for a self-sufficient sub! 
[QUOTE=Xema]
For most subs most of the time, navigation is done inertially and other craft are detected by listening, not by transmitting sonar signals.
[/QUOTE]
Listening is sonar. It’s passive sonar, as opposed to the “ping” of active sonar that you see on WWII sub movies all the time. And it’s more expensive, more technically demanding, and less useful for avoiding rocks than active sonar - rocks tend to be quiet. So a nice active sonar set would probably be useful for this case.
And here’s a company that may be able to provide just what you need.
[QUOTE=Xema]
For most subs most of the time, navigation is done inertially and other craft are detected by listening, not by transmitting sonar signals.
[/QUOTE]
[QUOTE=muldoonthief]
Listening is sonar. It’s passive sonar, as opposed to the “ping” of active sonar that you see on WWII sub movies all the time. And it’s more expensive, more technically demanding, and less useful for avoiding rocks than active sonar - rocks tend to be quiet. So a nice active sonar set would probably be useful for this case.
And here’s a company that may be able to provide just what you need.
[/QUOTE]
Muldoonthief is correct; listening is sonar, that is passive sonar. However, passive sonar in and of itself is not tremendously useful unless the information is fed into a target tracking system (referred to as a fire control system in the U.S. Navy). This can even be done manually, but it is a fairly manpower intensive process.
Active sonar would be more useful, but is likely to be fairly short-range for target detection unless very high sound-pressure levels are used, and then you’re likely to have issues with harming whales, etc.
Active sonar is very helpful for use as a fathometer (to determine depth).
With respect to navigation, realize that inertial navigation systems used by modern submarines are not exactly off-the-shelf items. IIRC, each of the three independent inertial units (plus two spares) cost in the seven-figure range.
[QUOTE=Xema]
I’d say that equipment to make fresh water is considered close to essential on any vessel that plans to be at sea for more than a week or so, let alone a submarine gone for months. Such devices certainly consume power, though not vast amounts. They are common on offshore yachts, and not actually all that expensive (examples here).
[/QUOTE]
It depends on how you make it. Distillation units are very power-intensive systems, which is not a problem with nuclear submarines. Reverse osmosis systems are less power-intensive, but typically make much less fresh water.