I'm trying to store some homemade organic red wine in bottles. For how long can it be stored without using any chemicals?
Preservatives generally aren't a part of winemaking, at home or commercially. If you are referring to sulfites, these are added to kill the yeast to stop fermentation at a specific point, not to preserve the wine per se.
Most sulfites dissipate, break down, or precipitate out within about 24 hours after application. They are not preservatives because they will not protect a wine from contamination weeks or months after application. Vinegar, salt, and sodium benzoate are food preservatives because they inhibit microorganism growth long after application.
Some people can have an allergic reaction to sulfites even in extremely small amounts, so it is best for these people to seek out wines that have not used sulfites at all.
There are other ways to stop fermentation besides adding sulfites. I've heard of chilling the wine in the vat to cause the suspended yeast to precipitate to the bottom, then drawing off the clarified wine from the top. Pasteurization is also a possibility, though I suspect the heat will alter the flavor profile of the wine.
When making your own wine, it's not technically necessary to do anything to remove the yeast from the wine before bottling. However, the risk of leaving live yeast in the bottle is that it could continue to ferment which will lower the sugar content of the wine (changing it's flavor) and increase CO2 pressure inside the bottle. This could cause the bottle to explode or pop its cork, making a mess of your wine cellar. If it doesn't pop the cork, the extra CO2 will make the wine fizzy when poured. If you want a sparkling cider or fruit wine this is great but if you want a smooth deep red wine it's bad.
The best way to limit the risk of blown corks is to minimize the amount of yeast that gets into the bottle (by cooling or filtering, or sulfiting) and then store the bottled wine in a very cool cellar with a very constant temperature. If your storage area can change temp by 5 or 10 degrees seasonally, the live dormant yeast will reactivate at warmer temps and blow your corks.
I've had corks blow out on me because I don't always sulfite before bottling, and my "cellar" is really just a dirt floor crawlspace under the house that while cool all year, does experience seasonal temperature fluxuations that you wouldn't see in a sealed cellar vault.
All wines will change in the bottle over time, but each wine matures at a different rate and in different ways depending on the chemistry in the bottle and how it is stored (temperature swings, etc). One of the things that happens in aging is tanins break down. Tanins give wine a sharp tang and some bitterness. A young wine will have a harsh flavor, but that same wine left to age a few years will become much more mellow as the tanins break down.
White wines and fruit wines generally don't improve with age. You drink them the first or second year after bottling. Cider and beer benefit from aging at least a few months in the bottle to "take the edge off" but are usually consumed within the first year or so of bottling. Red wines, because they contain so much tanin, often improve with age on a time scale of years to decades.
I know a fellow who seeks out "cheap" red wines with a particular flavor profile that he will age for 5, 10, 15 years before sharing with friends. He buys several cases at $5 a bottle but after years of aging he enjoys a wine with the depth and complexity of a $50 bottle.
As long as wine is kept sealed, cool, and dark it shouldn't ever go "bad" in the sense of becoming toxic. It will eventually reach its peak flavor profile and then gradually become less palatable over time.
Wine that has lost its seal (cork dried out because of improper storage, cork compromised by mold because of improper storage) will most likely "just" turn to vinegar but could also be toxic. If unsure, discard.
I don't think that you can actually put a specific time table on how long Organic wine can be stored. It will depend greatly on the style of the wine. Wine really doesn't spoil per say but it becomes less enjoyable to drink as it passes it's prime. As with most wines and beers there is definitely a point of diminishing returns when it comes to aging and about the only way to determine when that is would be to taste it. You can make some obvious assumptions about how long based on the style, the stronger bolder wine will age longer and the lighter more subtle wines will deteriorate faster. Other than the basics you will have to make a decision on a per batch basis. You can also expect that since there are no stabilizers or sulfites added to Organic wine that they will be more likely to hit their peak sooner than a treated wine.