I agree with most people that it would be nearly impossible without a Manhattan Project style commitment from the Government and I don't see that happening. That said, an interesting thought experiment would be to put a Falcon 9 second stage to 'sleep' on an up coming mission for a couple of weeks and then have it wake up and relight. If that works push it to a month. With a system like that you could Falcon Heavy to…
They've done extended, multiple relight tests of the second stage as part of their Air Force mission rating tests. I don't think it is anywhere near designed to stay usable after multiple days of staying cold, that is a completely different mission profile from what the second stage was designed to do, aka earth orbit missions. I would expect the stored cryogenics would not take well to it.
The soviets did experiments with a blok d upper stage that was also kerosene/LOX and were able to push it to a couple days, maybe up to a single week.
So really multiple weeks are not doable due to LOX alone withou turning your already mass limitted upper stage to an advanced orbital propelant depot.
Also longer term you will start hitting also other issues, such as battery power (might need to add solar panels), stuff getting too cold/hot, long term exposure of electronics to cosmic radiation, lack of ulage fuel, etc.