Assuming you don't care about time, effort, etc., what would be the
simplest, most naive approach to compiling all of US law into a single
body of work? Is it even theoretically possible?
The hard part is not statutory law, which is consolidated in the United States Code, or regulatory law, which is consolidated in the Code of Federal Regulations. The hard part is case law. Case law interpreting each section the United States Code (or comparable state statutory codifications of statute law) are accompanied by what are known as "annotations" which briefing state the holdings of all legal cases interpreting a particular statutory section and usually briefly stating its holding.
There are two main existing efforts to present other case law in codified form.
The more complete and ideologically neutral of them is the West Digest System. West hires lawyers to summarize every legal holding in every legal case available to it and to assign each such legal holding to a place on its subject matter outline it has created of all U.S. law. If you want to know about, for example, the constitutionality of police action shooting someone who is fleeing a law enforcement officer, there is a key number somewhere in the West Digest system that talks about that.
The other approach is an ongoing effort by the American Law Institute called the Restatements, of various areas of law governed by common law. The Restatement seek to summarize various areas of the common law (e.g. contacts, torts, judgments, restitution, property), in a statute-like form with commentary that courts often resort to and adopt in situations where their jurisdiction has not previously addressed an issue of law arising under the common law. Some of the Restatements have been prepared multiple times (with a first, second or even third restatement of a particular subject area) where the law has evolved significantly over time from what it was when the previous restatement was written.
Restatements are written by committees led by a distinguished law professor in the particular field called a "reporter" and unlike the West Digest System sacrifice being fully and accurately descriptive, in favor of articulating a single coherent theory of an area of law that theoretically represents a majority position of the states on that body of the common law, something that is more art than science. Often the reporter with the support of the relevant committee will adopt the "better rule" or trending rule rather than the actual majority position of the states on a common law issue. So, they can be not just oversimplifying, but also prescriptive in a legislative sense (something that is overly done when a "model statute" often in a new or evolving area of law is proposed).
Both of these code-like summaries of the case law are protected by copyright and not available to the public free of charge despite their relevance in how the statutes are interpreted.
So, in short, your project has really already been accomplished at great time and effort.
My initial thought was to find all of the relevant information (all
the many volumes and documents), but even this seems difficult. How
does anyone (citizens, lawyers, judges, other government officials,
etc.) even verify that the actions they've taken are legal when the
potential applicable laws are so vastly separated from each other
being in entirely different series of documents written in format of
People hire lawyers who are both trained regarding the broad outlines of what the law says in law school, so they can "spot issues" in a fact pattern that need to be researched and have answer determined, and who can find those answer either with computerized searches of primary sources, or with tools like annotated statutes and the West Digest System and a variety of other secondary sources (e.g. treatises on particular areas of the law and law review articles), to identify the relevant law.
This turns out to be much easier than it seems naively, because the vast majority of statutory law and regulations are relevant to only a very modest number of people and is subject to only very slight case law interpretation. Also, codified statutes and regulations are well organized enough that one can usually figure out which parts are likely to be relevant merely by looking at the table of contents and you can do even better if from experience you know where a few regularly important portions whose location in the table of contents is not facially obvious or whose location is counterintuitive from the face of the table of contents is known.
Most statutory and regulatory language pertain either to the organization of the government bureaucracy or to the regulation is enterprises in highly regulated industries that the vast majority of people have no involvement in and hence no need to know about. For example, there are vast numbers of pages of regulations governing topics such as alcohol labeling, municipal water quality determination, and life insurance company book keeping. But, if you are not a manufacturer of alcoholic beverages, a municipal water treatment plan, or a life insurance company accountant, you don't need to know about any of those things.
Measured by pages of annotated statutes, maybe 10% of statutes are pertinent to private law (i.e. non-criminal law governing interactions between non-governmental actors outside a regulatory scheme), to criminal and traffic laws, and to the portions of public law pertinent to ordinary individuals (e.g. the portions of tax law applicable to individuals and businesses who are not part of specially regulated industries or transactions). Probably something like 90% of the common law principles applicable in private law cases are taught in the first year of law school.
Case law is roughly 50% criminal and criminal procedure, 35% non-criminal common law and civil procedure, and 15% non-criminal statutory and regulatory and administrative procedure (with about two-thirds involving the small subset of statutes that pertain to private law as opposed to criminal and administrative law). Within any of those categories, maybe about half plus or minus is procedural and half plus or minus is substantive law. Even within non-criminal, non-private law, non-procedural statutes, the case law concerns a very small subset of overall statutory law. For example, in the case of the corporate code or the statutory law of environmental regulations, there maybe 5% or fewer of the statutory sections are litigated on any regular basis.
There are also distinctions between states, but the broad outlines are often generally similar in all states and where they differ it is frequently the case that there are only two to four rules of all in place in all of the states combined with one majority or plurality rule and one to three minority rules. States are much less original that the federal system of government permits them to be.
Also, a lot of common law and criminal law merely codifies common sense and cultural norms that already have wide intuitive understanding and compliance by people who are well behaved even in the absence of actually notice of the precise legal theory that compels particular conduct, and parts that aren't as intuitive and common sensical (e.g. traffic laws or professional regulation) are often subject to programs that require a license that is conditioned on a training program in the relevant laws before the license is issued.