What will happen to my owned games if Steam were to close?
TL;DR Summary: Valve has no legal obligation to do anything. Many gamers believe that Valve has said in the past that it would make the games available offline, but no definitive proof is public right now. It should be possible to backup all your games with Steam in offline mode, but may take a lot of space. Other options aren't great since most games require at least CD-activation.
The answer to this question is debated. Here's what we know:
Many gamers claim to remember (myself included) Valve making public statements in the early days of Steam (10+ years ago) that it would ensure that if the service ever went down gamers would be able to take their games offline. However, as of this post (Feb 2016), nobody can seem to find any such statements online anymore (if they did, in fact, once exist as we believe).
Some gamers claim to have asked Valve customer support the question directly and claim that the response confirmed that Valve has a failover/transition plan in the unlikely event of Steam shutting down for its users to get their games. Confirmation of customer support discussions is difficult. Some gamers have posted screenshots claiming to be of the response from customer support. However, it's impossible to verify the accuracy of such claims from gamers. Further, even if customer support did actually respond as shown in the screenshot, it's not clear that that answer will be valid indefinitely (or at all). It isn't considered a "public statement" persay and may have little legal validity either way.
The Steam Subscriber Agreement that all Steam users must agree to states clearly that the games are "licensed" not "sold" and that the Steam service constitutes a "subscription". Further, it disclaims any obligation to make the service available at all and may terminate the service at any time. All of these terms are standard for games-related EULAs (in fact, the Steam terms are far less ridiculous than many other EULAs) and are designed to protect Valve. The non-legalese interpretation is straightforward: Valve is under zero legal obligation to do anything for gamers should they decided to shut down Steam. Period. The Steam Subscriber Agreement is the only "public" statement that is presently available, and, regardless of the previous, almost certainly constitutes the whole of the legal obligation of Valve.
Finally, many gamers (myself included) are of the opinion that, independent of legal obligation, Valve has a history of being an (mostly) ethical company (certainly compared to, say, EA) and would probably continue to act accordingly in the unlikely event that Steam was shut down, making games available for all its users over some sort of transition period. This is, of course, speculation and cannot be backed up by any verifiable public statements right now. Users that are seriously concerned about this have three options:
Treat all Steam game purchases as potentially temporary and adjust the amount you're willing to spend on them accordingly.
Contact Steam customer service yourself and ask and keep a record of their response. Such a response may or may not be useful in the future from a legal standpoint given the Steam Subscriber Agreement (and depending on legal jurisdiction), but may provide some comfort to concerned users in terms of Valve's goodwill intentions.
In theory, it should be possible to install Steam, install ALL the games in your library (which could take up a very large amount of hard drive space, depending how many games you have), set Steam in offline mode, and then take a snapshot of the install (many possible options; one is to just clone the whole hard drive; another is to use the Steam backup feature) to preserve access to all offline games (see below for online games).
Beyond that, as most PC gaming has moved towards heavy online dependency, even with single-player games, the non-Steam option of purchasing physical discs (CDs, DVDs, etc.) with games on them that have no online activation, etc., is extremely difficult. Most such games require, at the very least, a CD-key activation that depends on a game company keeping a server running somewhere that could go away anytime. This issue has arisen in the past and some authors have argued strongly against such dependencies for consumers. The issue of gamers being dependent on the survival of companies and their goodwill is an industry-wide issue and not specific to Valve or Steam.
Multiplayer games present another challenge because the nature of the games themselves depends on not just individual users' access to the game software, but also on access to the multiplayer server software. Valve has made some of its multiplayer server software (such as those related to HL2, Counter-Strike, TF2, etc.) available for users to download and run as private servers, so users can already play without dependence on Valve multiplayer servers. However, that's not generally the case with all Steam games as many other companies that sell their games through Steam maintain their own multiplayer game servers and don't allow individuals to set up private servers. What would happen if Steam shut down in those cases is not clear and would depend on each of the many individual such companies' choices. One would hope they would simply release the servers in addition to the game software as part of a transition process, but since Valve doesn't have control over those servers nor those servers' software, it likely wouldn't be up to them (unless it's specified in the company-to-company contract that other companies must agree to in order to put their games on Steam, but no such contract(s) is(are) known to be publicly visible.
Note: This answer was edited and cleaned up to address modifications to the original question and issues raised in comments.
I would like to add to existing answers by also saying that Steam is the delivery method and storefront, for this they take a cut. But if they are to go under, Valve can not just decide to give you the games as the rights holder is still the publisher.
Publisher may choose to sell the game through Steam, using it as a storefront to attract market and gain sales. They may also use Steam services as part of their game.
For this, they agree to let Valve take a cut on the sale. But the game code itself merely resides with Valve, not the rights to it. It is not Valve choosing to put a game on Steam, it is the publisher USING Steam as distribution method.
So if Steam goes down, Valve may want to give you a means of accessing your game, but the publisher may decline.
Valve has no say in that, as they run a store, and they do not own the files they distribute, nor the IPs. The publisher retains the right to their product. So if Steam stops selling them the distribution service, the publisher has no obligation to give you a new copy on a different medium, unless THEY (not Valve) say so when you buy.