There are two matters at hand to answer this. The first is pertaining to GOG specifically. When you buy a game from GOG, it will remain in your account indefinitely including if the publisher removes the game from the store. Once you own a game on GOG, you own it permanently.

The second, is what the licensing terms (EULA) are for the game as provided by the publisher of the game itself. That varies greatly from game to game and there is generally always a EULA file included with every game, whether it is EULA.txt, README.txt or some other file name, etc. What the specific terms are in that EULA will vary wildly from game to game and publisher to publisher, and that determines what rights they are granting you to their game intellectual property under copyright law.

As far as GOG is concerned, you can download your games from GOG whenever you want as long as they're still in business and their servers are still running. The games being DRM-free, the idea is that you have the right and the ability to keep backup copies of your game and bonus content for your own personal usage. GOG's intent is to continue to provide access to the downloadable files indefinitely of course, but it is up to you personally to make backup copies for yourself in order to account for unforeseeable circumstances. For example if GOG were to go out of business or something, then there would likely be a limited time window where people would have to download backups of their games until the company shut down their servers. That is just theoretical however, and with GOG growing fast in the gaming industry such a situation seems unlikely and low-risk. Keep in mind too though that Desura, and ShinyLoot gaming retailers both went out of business earlier this year, unable to compete with Steam, GOG and other companies and turn a solid profit, and many people who were not following things closely may have lost out on their game catalogue at those stores if they had not redeemed and downloaded all of their games on time.

But what your actual legal rights are for a specific video game are not determined by GOG.com, but rather by the individual game companies, documented in their EULAs. In order to know what your specific legal rights are for an individual game you would need to read these license documents for each and every game individually, and also have some knowledge of how copyright laws work in your own country. Most countries in the world follow the Berne Convention for copyright law, however many countries have additional laws that supplement that as well. I know Germany does for example.

But there are two sides of this too. There is the pure legal side of things, which is what the EULAs say and whether such agreements are actually legal and would hold up in a court of law. But then there is the fact that for the most part those legal agreements do not really matter to the individual person because if you burn a copy of a game on DVD for yourself, whether or not the license permits this - how the hell would anyone ever know what you did? Unless you're pirating games and putting them on the Internet or doing something blatantly illegal that could land you in a court of law over copyright infringement, making personal copies of games for yourself for backup purposes, whether the legal agreement permits it or not, whether the laws of your country permit it or not - really doesn't matter if you're not doing anything with it that would ever be known to anyone else. It's like buying an audio CD and making a mixed CD of tunes or an mixed-MP3 playlist on your media player - is it legal? Maybe, maybe not depending on the country's laws and other factors, but who actually cares? It's not like there are video cameras in your house monitoring you to see if you're violating the EULA of a video game by burning it on DVD.

Nobody cares really. :)

The bottom line is that the idea is to download copies of your games so you own backups that you control which you do not need to rely on GOG existing any more, and if they go out of business, then you have a copy of your game permanently. Whether you store them on hard disks, DVD or chiseled in clay tablets is up to you, and whether it is technically legal in a court of law really doesn't matter even the slightest bit unless you actually find yourself in a court of law - which is highly unlikely unless you're purposefully doing something to make the law come and look at you, which you most likely are not going to be doing.

In other words, you may be thinking about it too much. :)