The short answer to your question is that no, a Roman citizen sitting quietly at home minding his own business could not lawfully be enslaved. It was a major crime to kidnap a citizen and sell him as a slave. The citizen was never actually a slave-at-law; the citizen was the victim of crime and did not need to have his rights "restored" because he had never lost his citizenship.

The details of the Roman Law of Slavery changed greatly from the terms of the Laws of the Twelve Tables, through the Late Republic, then under the empire and then in the aftermath of the rule of Constantine the Great. However as a general matter, Roman law provided only two ways in which a person could lawfully be enslaved: (1) status at birth; and (2) free status changed through state action. Broadly speaking, the children of enslaved females took the status of their mother. Thus the children of a slave woman were slaves while the children of a free woman were free. However, there are enough exceptions to that rule--and so many permutations of the exceptions--that W.W. Buckland devoted about three hundred pages to the discussion of means and methods of enslavement at birth in his book, The Roman Law of Slavery (1908). For the purposes of this discussion, the general rule is probably enough detail.

State action falls into several major categories including: (1) enslavement pursuant to war; (2) enslavement for debt; and (3) enslavement upon criminal conviction. A captive taken in war could be lawfully enslaved as long as Rome was formally at war with the captive's nation or state or city. If there was no formal state of war, then a person who was seized and enslaved was the victim of piracy and kidnapping--major crimes. A person could be enslaved pursuant to a legal action for debt, but could not simply be seized by the creditor and sold. Self-help was not a legal option. There had to be a judicial proceeding for debt slavery to be lawful. The same is true for enslavement based on a criminal act: the perpetrator had to be adjudged guilty and sentenced to death or to serve as a slave. A death sentence could be commuted if the person was remanded to the arena to serve as a gladiator or if he or she was condemned to the beasts. (There are some other exceptions to this which fill several chapters in Buckland's work.)

The minor children taken and enslaved in HBO's Rome were the victims of a crime. If they could have contacted a praetor or magistrate, they could have appealed their enslavement and been immediately released. If a free man identified them to a magistrate, they could have been immediately released. However, in the show they were clearly the victims of kidnapping by a crime lord and they never had any opportunity to plead that they were Roman citizens. Vorenus, as their father had the right to rescue them and seek damages for the diminution of their worth. Vorenus, being Vorenus, resorted to self-help and executed their kidnapper and eventually freed the children--and murdered the slaver in the process. He was not entitled to take those steps. If the slaver knowingly used Roman citizens as slaves, that was a crime and he might have been sentenced to death as a result. However, if he unknowingly and in good faith bought citizens and used them as slaves, that was a matter for civil damages payable to Vorenus as the father of the children. A buyer who made a good faith purchase of Roman citizens had a cause of action against the criminal who sold the citizens to him. But Vorenus acted outside the law, took his vengeance and rescued the children--who had never legally been slaves no matter how they were used by a bunch of criminals. It's doubtful that any action could have been taken successfully against Vorenus for these acts because he was in the military (which had its own elaborate set of laws and protections) and he arguably was Marc Anthony's client.

Maximus' problems in Gladiator may or may not have been state action. If Commodus, after assuming the imperial mantle, condemned Max to death or to the area or to the Baetican mines--that would have been state action and Max's enslavement would have been lawful. However, there is some implication in the movie that Max was merely sleeping by the side of the road and was seized by kidnappers who sold him. Now this presents a nice legal problem. If this scenario resulted in Max' enslavement, this was a crime and he was a victim. However, he seems never to have protested his enslavement. At a certain point, an unlawfully enslaved citizen who failed to timely challenge the enslavement lost the right to complain because he was deemed to have acquiesced in the result. He thus lost his citizenship and was indeed a slave. (Timeliness for the purposes of this rule required an actual opportunity to complain. If the victim had no opportunity, then the clock didn't start running against him.) So Max probably really was a slave by the time Commodus murdered him in the arena.

If you are interested in the topic, I suggest you take a look at Alan Watson's "Roman Slave Law," (1987). It's a good summary of the subject and doesn't rely on highly technical Latin legal terminology as does Buckland.