According to the Federalist Papers, no, the general welfare clause does not enable the government to do whatever the heck it wants:
It has been urged and echoed that the power “to lay and collect taxes, duties, imposts, and excises, to pay the debts, and provide for the common defense and general welfare of the United States,” amounts to an unlimited commission to exercise every power which may be alleged to be necessary for the common defense or general welfare. No stronger proof could be given of the distress under which these writers labor for objections, than their stooping to such a misconstruction.
Had no other enumeration or definition of the powers of the Congress been found in the Constitution than the general expressions just cited, the authors of the objection might have had some color for it; though it would have been difficult to find a reason for so awkward a form of describing an authority to legislate in all possible cases. A power to destroy the freedom of the press, the trial by jury, or even to regulate the course of descents, or the forms of conveyances, must be very singularly expressed by the terms “to raise money for the general welfare.”
But what color can the objection have, when a specification of the objects alluded to by these general terms immediately follows and is not even separated by a longer pause than a semicolon? If the different parts of the same instrument ought to be so expounded as to give meaning to every part which will bear it, shall one part of the same sentence be excluded altogether from a share in the meaning; and shall the more doubtful and indefinite terms be retained in their full extent, and the clear and precise expressions be denied any signification whatsoever? For what purpose could the enumeration of particular powers be inserted, if these and all others were meant to be included in the preceding general power? Nothing is more natural nor common than first to use a general phrase, and then to explain and qualify it by a recital of particulars. But the idea of an enumeration of particulars which neither explain nor qualify the general meaning, and can have no other effect than to confound and mislead, is an absurdity Madison, Number 41
From these observations this conclusion results: that the trial by jury in civil cases would not be abolished; and that the use attempted to be made of the maxims which have been quoted is contrary to reason and common sense, and therefore not admissible. Even if these maxims had a precise technical sense, corresponding with the ideas of those who employ them upon the present occasion, which, however, is not the case, they would still be inapplicable to a constitution of government. In relation to such a subject, the natural and obvious sense of its provisions, apart from any technical rules, is the true criterion of construction.
Having now seen that the maxims relied upon will not bear the use made of them, let us endeavor to ascertain their proper use and true meaning. This will be best done by examples. The plan of the convention declares that the power of Congress, or, in other words, of the national legislature, shall extend to certain enumerated cases. This specification of particulars evidently excludes all pretension to a general legislative authority, because an affirmative grant of special powers would be absurd as well as useless if a general authority was intended.
In like manner the judicial authority of the federal judicatures is declared by the Constitution to comprehend certain cases particularly specified. The expression of those cases marks the precise limits beyond which the federal courts cannot extend their jurisdiction, because the objects of their cognizance being enumerated, the specification would be nugatory if it did not exclude all ideas of more extensive authority.
-Hamilton, Number 83