In addition to what John provided, it's also worth noting that given the scale of tens of thousands of years, the stars will actually move. This is clearly shown in a Wikipedia article. As can be show, Alpha Centauri will only be 3 Light Years away in about 30,000 years.
Okay, so the fastest mission I've heard of using nearly obtainable technology is the method detailed in this question, graphic provided from XKCD. It results in a speed of 20 au/year, or about 95 km / second. Give that speed, how long would it take to go to another star? Well, that speed is equivalent to 1 light year per 3000 years, give or take. Thus, given the fact that Alpha Centauri is getting closer, I'd say around 12,000 years or so would be possible.
(image license: CC BY-NC 2.5)
What are the challenges associated with this? I'm going to include a very unscientific list, in my thoughts about what order they should be.
- Power- Keeping the spacecraft powered in the void.
- Longevity- It's hard to make something that still works after 50 years, let alone 12,000... Furthermore, space is a difficult environment by it self, including a high amount of radiation, making it even more difficult.
- Future Faster Space Probe- A huge expense could be spent to launch this spacecraft, and have a spacecraft 100 years later that travels 10% faster reach the system well before. Or it could even be much faster, leading to even more discrepancies.
- Communications- There is a significant link margin required to make this work.
- Light Speed Delay- Being so far away, the spacecraft would need to decide 100% of what it did, there would be no way to take a few pictures and command a new stack
- Objects of Interest- Some work would need to be done to find objects of interest to photograph, such as planets
- Course correction- This should be easy enough to accomplish, but would require keeping fuel in the void for a long period of time. Fuel usually has requirements about what temperature to store it at, which might be difficult for such a long journey.
- Technology not getting out of date- It's hard to make something still compatible with 50 years of technology, let alone 10,000+..
- There's always a fear with such long term items that someone a thousand years later will find a better method to send the probe that only takes 8,000 years, thus making the probe you sent out entirely useless, as it will arrive 2,000 years too late.
And there are no doubt quite a few other very serious issues. Bottom line is, I think the following minimum technology is required to make this feasible.
- A way to make the journey in less than 100 years.
- Careful observation of the star system to ensure that targets of interest are identified.
- Perfecting Laser Communications, or similar, to allow for rapid far communications to occur.