To answer the question that OP asked ("Can California ban its citizens from traveling to other states?"), which I should note is a purely hypothetical question (i.e. California has not actually attempted to do this, as explained in other answers), we can look to the Constitution. Article IV, Section 2 begins as follows:
The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States.
This is a bit vague, but multiple courts have held, over an extended period of time, that this clause prohibits states from interfering with the right of US citizens to travel between states. In Corfield v. Coryell (1823), a federal district court held as follows:
The inquiry is, what are the privileges and immunities of citizens in the several states? [...]
The right of a citizen of one state to pass through, or to reside in any other state, for purposes of trade, agriculture, professional pursuits, or otherwise; [...] These, and many others which might be mentioned, are, strictly speaking, privileges and immunities, and the enjoyment of them by the citizens of each state, in every other state, was manifestly calculated (to use the expressions of the preamble of the corresponding provision in the old articles of confederation) "the better to secure and perpetuate mutual friendship and intercourse among the people of the different states of the Union."
(Significant amounts of text removed because the writing is quite verbose.)
Later, in the Slaughter-House Cases (1873), the Supreme Court of the United States cited this case favorably (while interpreting a different but similarly-worded provision of the Fourteenth Amendment). More recently in Zobel v. Williams (1982), Justice Sandra Day O'Connor wrote this in her concurrence:
Article IV's Privileges and Immunities Clause has enjoyed a long association with the rights to travel and migrate interstate.
The Clause derives from Art. IV of the Articles of Confederation. The latter expressly recognized a right of "free ingress and regress to and from any other State," in addition to guaranteeing "the free inhabitants of each of these states . . . [the] privileges and immunities of free citizens in the several States." While the Framers of our Constitution omitted the reference to "free ingress and regress," they retained the general guaranty of "privileges and immunities." Charles Pinckney, who drafted the current version of Art. IV, told the Convention that this Article was "formed exactly upon the principles of the 4th article of the present Confederation." 3 M. Farrand, Records of the Federal Convention of 1787, p. 112 (1934). Commentators, therefore, have assumed that the Framers omitted the express guaranty merely because it was redundant, not because they wished to excise the right from the Constitution.
She then goes on to favorably cite Corfield v. Coryell and several other cases before finally concluding that:
History, therefore, supports assessment of Alaska's scheme, as well as other infringements of the right to travel, under the Privileges and Immunities Clause.
While no justices directly joined her concurrence, Justice William J. Brennan wrote this in his concurrence:
I agree with JUSTICE O'CONNOR that these more fundamental defects in the Alaska dividend distribution law are, in part, reflected in what has come to be called the "right to travel."
He then discusses whether the right comes from the Privileges and Immunities clause or one of several other clauses, before ultimately concluding that it does not matter because the right clearly exists:
In any event, in light of the unquestioned historic recognition of the principle of free interstate migration, and of its role in the development of the Nation, we need not feel impelled to "ascribe the source of this right to travel interstate to a particular constitutional provision." Shapiro v. Thompson, 394 U. S. 618, 394 U. S. 630 (1969)
Brennan's concurrence was joined by three justices. Including Justice O'Connor, a majority supported the conclusion that the right to travel is guaranteed by the Constitution, so this is now binding precedent on lower courts.
TL;DR: No, California is not empowered to restrict the travel of its citizens in the manner OP is describing. However, as other answers have pointed out, California is not attempting to do that in the first place.