It is probably illegal to download the games from this site even if you already bought a copy of them. Most manufacturers of video games do not give the purchaser the right to download additional copies.
Even if the products are no longer being produced, they still will be subject to copyright infringement, which is based on the life of the author plus 70 years from the author's death.
There is a simple rule of thumb for music, video games, movies, music videos, books, articles, etc.---if you can buy a copy on Itunes, Amazon or other such sites, then you cannot legally download it for free. No exceptions, no complications, no excuses. The people who create these works depend upon payments from consumers in order to generate sufficient revenue to permit them to do their work and make a decent living. People who download for free essentially are engaging in theft. It may not be the crime of the century--but it is still illegal.
When a web-site operator attempts to induce you to use their site to download video games and other such works, it is quite possible that the web site operator will falsely claim that use of the site for this purpose is legal. You should be suspicious whenever a web-site operator makes the claim that you can download video games, music, books, movies or other works for free. Even if the web-site operates outside of the United States, your presence in North Carolina means that your downloading of a game from the web site probably violates US copyright law. This is true even if the web site-itself, is operating legally.
A lot of people still believe that the internet is the wild west--and that all content on the internet should be free. This is simply not true. The Digital Milleneium Copyright Act prohibits people from engaging in efforts to circumvent copyright protections. Each act of infringement, if willful, can subject the actor to damages of up to $150,000. The major video game producers, record labels, movie studies and publishers are beginning to crack down on illegal downloads, and the courts have been willing to impose substantial liability on infringers. The safe, ethical, and legal approach is to pay for your video games.
One thing you have to consider is where these types of websites are based, and why they are based there. The one you have mentioned appears to be based out of the Bahamas. I believe another one called Pirate Bay is based out of Sweden or somewhere in that area. These types of websites don't operate out of the U.S. because we have strong federal copyright protection for music, visual arts, and software. Their activity may likely be legal wherever they are located, but that doesn't mean it is legal for you to download. You're in North Carolina, and that changes everything for you.
I can't say with certainty whether the website you mention has indeed obtained copyright ownership or a proper licensing agreement to do whatever and all that they are offering. (I'm trying to give them the benefit of the doubt here.) I can tell you that it would be no small feat for them to do that though. You're relying on their statement to save your own tail if you ever got sued, and that is not really a convincing story to a judge or jury.
Also, an FYI, the copyrights to games from the 80's and 90's are still well in effect and won't expire until you're retired. Whether or not the games are still being produced and sold at retailers is irrelevant. I would be very skeptical to believe that they have obtained proper licensing agreements to allow anyone to download the game. There's a reason they operate outside the U.S.
Disclaimer: In no way is this information considered a legal advice, but is only a statement of the law. This information does not form a client-attorney relationship.
Only the copyright owner can make copies of their work. If your original purchase of the game included the right to upload the game on more than 1 device (like iTunes' terms of sale provide, with the knowledge that people often own several iPods), then you can install the game on as many computers as the games terms of sale provide. If it limits you to 1 device, however, they intend for you to buy as many copies of the game as you want per device, and to buy updated versions of the games, if available, or new games, if not.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.