Is there any good scientific reason someone couldn't be kept in hibernation, cryo-sleep, suspended animation or what have you indefinitely, or at least up to a few thousand years? If not, what would the challenges be in accomplishing this? Would it be feasible with technology developed at any point in the 21st century?
The biggest issue is that we really don't know what it means to be living. We've never successfully resuscitated an individual that was frozen, so we don't even know if freezing is an effective way to extend one's lifespan at all! Current technological limits are mostly related to ice crystal formation (both during freezing and while frozen). Future issues could include proteins that are denatured during the freezing process. Once those are solved, we can raise the more interesting question of whether you are the same person when you come back out!
Presuming you can actually freeze a human and still have them "living," the only limit would be how long you could keep them cold. Likely the colder the better. Thermal noise will eventually reorganize molecules, and we have plenty of molecules that we really don't want reorganized. Freezing them very very cold would slow thermal movement and increase longevity.
I wouldn't consider a person being "frozen" as "still alive." If you could look at their cells with a microscope, there would be no movement, no uptake of ATP or expulsion of waste. No leukocytes circulating around, cleaning up bacteria or other junk. Brain cells also wouldn't fire, so there would be zero brain activity. For all intents and purposes, they are dead - very dead.
It seems exciting, this possibility of "thawing" and coming back to life, but it isn't very realistic for a human. There are only a few creatures which can survive being frozen (select fish and frogs that I know of.) But the fish essentially hibernate with anti-freeze in their bodies, and the frogs pump their bodies full of glucose, will die if the temperature drops too low, then have to repair damaged cells when thawed. Neither can withstand being frozen completely solid, nor for very long periods.
@Tryss has a good point about the sustainability of suspended animation. If a body were cryo-cooled using liquid nitrogen, then where is this constant supply coming from for 1000 years? If it's being extracted from the atmosphere, where is the power coming from to run this machinery? Will it ever break down? How many back-ups are in place?
Another thing to consider is (nuclear) radiation. The environment and Earth itself is slightly radioactive, called background radiation. It's present at all times, day or night, and is caused by the decay of atoms all around us. In a healthy body, radiation can damage cells very much like a microscopic bullet. In a living body and with low exposure, the cell is repaired ~100% of the time. There is a slight chance that the radiation may damage a cell in a way in which it continues to live but changes slightly. If that happens, the body tries to isolate that cell to prevent it from multiplying. If the body fails at this and it multiples out of control, then we call this cancer.
Bottom line is, with zero automatic repairing of damaged cells over 1000 years, there would be little chance of survival due solely to cellular degredation from radiation exposure. The moment the person was "thawed", the body would be unable to cope with the repair demand. Even if they survived reanimation, the chance of cancer is greatly increased. The effect would be like they suddenly were exposed to all the radiation from those 1000 years instantly, i.e. radiation posioning. If in the United States, typical background exposure is ~6mSv/year (milli Sieverts per year.) 1Sv = 1Gy. According to this, exposure of 1 Sievert equates to a 5.5% chance "of eventually getting cancer." 0.006Sv * 1000yr = 6Sv, 6 * 5.5% = 33%. However, according to the acute exposure chart, being exposed to 6Gy of radiation has a 50-100% mortality rate even with medical care.
The only practical solution may be to imitate the processes used by frogs and fish - glucose, anti-freeze, NOT being frozen completely solid, and only "hibernating" for short periods, followed by a recuperation period to allow cells to regenerate. With heavy sedation, because it's going to hurt. Underground. With dozens of generators and backup generators. In a lead-lined vault to block out as much of the background radiation as possible. And then cells are still going to age over time; it won't be a true suspended animation. So extremely long timeframes are just not feasible.