They weren't able to. The current theory of the population crisis that lead to the land distribution program of the Gracchi brothers was due to people not registering with the government so they would not get called up for the widely unpopular Spanish Wars. See here by /u/XenophonTheAthenian. It's perhaps important to note that the Spanish Wars differed from many other wars fought by the Republic in that fighting was hard, losses were high, and loot was scarce, all the while they were not fighting against a foreign invader to defend Rome, but abroad in strange lands.

What era are you talking about for this as Rome lasted quite a long time? Also what do you mean by war fatigue? In a hard tangible sense; lack of money, soldiers, etc, or in a intangible sense, like the populace getting tired of war? “Soldiers and Silver” by Michael J. Taylor gives an answer for the second and third centuries BC and is the basis for this answer. It gives a comparative look at the monetary and manpower abilities of the five great Mediterranean powers during that time, Rome, Carthage, Macedonia (Antigonid dynasty), the Seleucids, and Ptolemies in Egypt. The short answer is that Rome was able to much more efficiently fund their military than the other powers which allowed them to be able to have many active theaters of war at once and mobilize the largest forces. This also allowed them to suffer massive losses and still continue fighting. They weren’t able to do so infinitely, Rome had its limits as well, but they were able to do so better than their great power competitors which allowed Rome to establish itself as the Mediterranean hegemon. The maximum mobilizations for the five powers during that time were as follows:

Rome: 185,000

Carthage: 165,000

Macedonia: 50,000

Seleucids: 80,000

Ptolemies: 80,000

While Rome had a strategic manpower advantage they were often at a tactical manpower parity or disadvantage for many of their key battles during this period. You would think that the relative finances would match or be close to the same ranking of this strategic manpowers, however they are not. Taylor estimates that the annual revenue for each power was the following annotated is millions of Attic drachmas:

Rome: 12

Carthage: 8 (Edit: This amount is for after the Second Punic War. It was probably closer to 16 million before the war. )

Macedonia: 8

Seleucids: 190

Ptolemies: 260s

Rome was able to pay for their military more efficiently. Taylor assesses that the pay for an average soldier in the Mediterranean was around a drachma. Rome was able to pay their soldiers less than 2 obols (there were 6 obols in a drachma), so less than a third of their competitors prices. Additionally, Roman armies contained a large portion, normally at least half or more, of troops from their Italian allies. These Allie’s paid and equipped their troops while Rome provides their rations. This lowered costs even more. Taking into account purchasing power parity the difference is even more. The maximum possible mobilized manpower based off of purchasing power and maximum recognizes were:

Rome: 215,000

Carthage: Not provided

Macedonia: 35,000

Seleucids: 140,000

Ptolemies: 160,000

You can see that Rome came close to the maximum manpower, Macedonia exceeded it (believed to be because they tapped into their cash reserves to fund it), while the Seleucids and Ptolemies were around half. Some of this was cultural. Rome was apparently very studious in their financial bookkeeping and austere when it came to non-military spending during that time.

The Seleucids and Ptolemies were the opposite and had huge non-military courtly costs such as feasts, processions, benefits to junior allies and so on. These two kingdoms needed to do this to ensure the loyalties of elites. They couldn’t just shut it off during war because they needed the elites even more during war. Macedonia is a bit of an outlier here as it was a kingdom but didn’t do this massive non-military spending. Part of that would be because they had the least people and money so they couldn’t spend it like that. The other is that they were Macedonian kings ruling over the kingdom of the Macedonians. The Seleucids and Ptolemies were Macedonian kings ruling over land they had conquered and had no ethnic ties to.

Another reason given is the difference between states that sought to extract value from subject people either through conscription or taxation. Each power did both but the relative weight between the two was what was important. Rome was the best at conscription and in the end won. Carthage went from heavy taxation to conscription during the Second Punic War and was able to field large armies that almost defeated Rome. Macedonia was conscription and was able to punch above its weight. The Seleucids and Ptolemies were both taxation that had to rely and a relatively small cadre of military settlers as the core of their armies, but this is a method that is relatively fragile. They could use their mass wealth to purchase mercenaries and they did so. However, there were only so many mercenaries and the other side (these two kingdom’s most common opponent was each other) was probably trying to do the same driving up prices and down supply. Some of the other powers could also be trying to purchase mercenaries for a different war at the same time or the mercenaries could be fighting a war for their home state. Often they had more money than available mercenaries.

Finally, Rome was a republic then which allowed for its societal infrastructures were more robust, penetrating, and resilient than those of the kingdoms. Legitimacy was given to the Roman government through its repeated votes. Carthage was also a republic and it gave Rome its toughest test by far. However, compared to Rome it appears that its structures weren’t as strong. Carthage probably had fewer citizens in Rome. Additionally, the ties to the subject manpower were generally either personal relationships with the Carthaginian elite (like weddings) or through harsher methods such as hostage taking. These methods are less resilient than the Roman methods.

EDIT: I should add that Taylor makes it clear time and again in his book that the absolute numbers listed above aren’t important, but rather the relative strength between them. Additionally the 8 million Attic drachmas for Carthage is after the Second Punic War and it was probably closer to 16 million before the war.