During the Republican period, the two consuls would each raise two levies of 4-5,000 men from the class of property owning men, for a total of 20,000 conscripts each year. This levy (the root of the word legion) would be joined by an equal number of troops from Rome's confederate Italian states, for a total of 40,000 men, divided into two consular armies of 20,000 men. These were not professional armies; they elected their officers, and would be disbanded at the conclusion of a campaign, to be raised again the next year under new Consuls.

By the time of the Second Punic War, Livy estimated that there were nearly 250,000 men of the required age and property qualifications in Roman territories, without even including Rome's Italian allies; Hannibal butchered them in droves, but by now Rome controlled almost all of Italy, which gave them vast manpower to draw on. Even after many of their allies defected to Hannibal following the slaughter at Cannae, Rome not only still had plenty of property owning men to draw on, plus overseas armies and garrisons, but were willing to ignore the property qualifications, opening up an even larger manpower pool to stem Hannibal.

/u/dandan_noodles provides a nice summary, but I would like to elaborate.

How could Rome constantly raise enormous armies despite not being that large of a country? The answer to this question is quite simple: the pool of available manpower accessible to the Romans is actually enormous. As stated by dandan_noodles, by the turn of the 4th into the 3rd century, the Romans had a pool of around 300,000 military age landowning men to draw upon for the army. However, considering the casualties the Romans suffered from 218-216 BC in their series of catastrophic defeats against Hannibal (130,000+), their manpower resources should have been at breaking point. However, this was not the case.

According to Polybius (2.23-24), whose numbers according to Southern and even number-crunchers like Brunt aren't far off, the Romans had nearly 800,000 (!) military age men to draw upon in the period of the Second Punic War, just in Italy. The aforementioned number of nearly 300,000 only includes the Romans themselves and their most proximal allies, the Latins. The rest of Italy made up the rest of these numbers. Admittedly with the defections in Italy after Cannae this number must have dropped, but even so, even in Rome's little corner of Italy, there were potentially 300,000 men that square up to Hannibal when properly equipped.

I have to emphasise that the Carthaginians also had a very large available pool of manpower, but Carthage was not financially or logistically Rome's equivalent. Even in the First Punic War Carthage didn't run out of men, but money to pay its huge mercenary armies (See Howarth). While control of the precious metals in Southern Spain no doubt helped, Carthage did not quite have Rome's capacity to keep huge numbers of men in the field for nearly indefinite periods of time, though of all the nations in the known world at the time it was probably Rome's closest competitor in this regard.

As for your second question, were the Roman armies of the mid Republic as professional as often portrayed? I am going to assume you mean their depiction in Hollywood, movies such as Spartacus, Gladiator, or shows like Rome by HBO. If your definition of "professional" is tight drill, total uniformity in dress and equipment, salutes and marching in time and all that, then no. The Roman armies of this period were not at all professional in this period or any period really. Their uniforms were never as same-ish as depicted in media, they wore different kinds of armour since they all equipped themselves, they didn't have such an obsession with the colour red (the Hastati wore white for example), they didn't have drill and they didn't march in time and they probably didn't salute.

If your definition of "professional" is, were they experts in the profession of arms, and were they highly capable and organised and immediately recognisable as Roman, and were they paid by the state to serve as soldiers, then yes, they were absolutely professional. Totally uniform training methods and dress and drill techniques are modern inventions, not something the ancient world ever had as we tend to understand it. You might consider the Roman infantry in the Empire to be more "professional" because they weren't disbanded after campaigns but stayed with their units for 20 years or more. But even in the middle Republic the Roman infantry were, in general, the best in the Mediterranean, and it took a very rare general like Hannibal to make them look bad through superior use of terrain, ambush and manoeuvre, and it certainly didn't work for him when he ran into a general of similar caliber at Zama. At Zama it was made abundantly clear how superior Roman infantry were in a straight-up fight to their equivalents from Gaul, Spain, Africa or anywhere else.

Sources:

Brunt, Italian Manpower, 1971.

Gary Forsythe, Critical History of Early Rome, 2005, 38.

Howarth from The Oxford Handbook of Warfare in the Classical World, Eds. Brian Campbell and Lawrence A. Tritle, 2013, 32-3, 42.

Pat Southern, The Roman Army, 2006, 49.