Here are the particulars:
What percentage of the population can act as a standing military long term during peacetime?
The primary role in peacetime is the protection of roads from bandits, and defense against barbarian raids.
I am primarily interested in finding out how many soldiers a population of 25,000 (soldiers included) can support.
Based on historical data, the adult population in a typical polity in the Middle Ages could support one fighting man for every 15 adults maximum.
As noted, fighting men didn't exclusively "fight", but were generally higher ranking Feudatories, who acted in administrative roles for those of higher feudal rank to whom they owed alligence. Since the web of relationships was often theoretical above a certain point (certainly too difficult to enforce effectively), much of Europe during the Middle Ages (or Japan during the period before the Tokugawa Shogunate or China during the Warring States periods) was divided into much smaller polities run by minor or "mid level" nobility. So in addition to being able to fight and supply a set amount of fighting material (coming with horses, armour and weapons), they also adjudicated disputes, supervised the harvesting of corn (wheat), the milling and distribution of flour, ensured markets ran (with their percentage of the cut) and so on.
The other reason the ratio of fighting men to the others included the high demands of time for training for war, and the resource bill for the man, armour, weapons, horses etc. This could be made up in part by a levy of the peasants to provide foot soldiers, hiring of mercenaries and the arrival of brigands who would be happy to serve for a share of the loot.
This relationship ended with the start of the Infantry Revolution, when simple to use weapons and tactics (crossbows, pike formations, pole arms) were introduced, allowing a mass of peasants or yeomen to take to the field and effectively fight against mounted knights, so if your setting is in the mid 1400's or beyond, then the percentage of effective armed manpower increases, although the amount of time that they could take to the field will decease (since they still need to carry out farming, crafts and other daily tasks to stay alive and prosperous).
There are two factors limiting the amount of man power a society can deploy, overall population and internal political cohesion. Of the two, the latter is the most important.
Medieval societies were societies built upon the military oppression and exploitation of the farmers (peasants.) They could never allow the peasants to become to well trained, armed or coordinated or they could defeat the aristocracy. They called up a few farmers as possible and forced them to provide their own weapons from converted agricultural tools and whatever armor they could improvise.
Farmers that got to skillful on the battlefield tended to end up mysteriously dead.
The European military warrior class the so-called "nobility" comprised roughly 5% of the population. Around 5% were clergy, artisans, traders etc and largely considered non-factors in warfare.
During the harvesting and planting seasons, essentially non of the farmer population could be called up out wrecking the nobles own fortunes worse than loosing a war would. In Winter time, armies could not travel so war occurred in the narrow summer months.
In theory, the nobles could require all peasant males in a certain age range, usually 17-50, to provide 40 days of military service a year during the summer. That would be roughly 15%-20% of the total population.But even in summer, no more than half the available man power could be done without on the farm so more like 7.5%-10%
However, save for the rare chance at plunder in a foreign land, peasants didn't get paid so they had little incentive to comply with their feudal obligations (which forced upon them by conquest) so getting enough men to show up was a constant struggle.
Neither did it help that well into early 1800s, military service of any duration beyond a couple of weeks was looked on as a death sentence. 2/3 of soldiers died of disease in unsanitary camps with little food and constant exposure. WWII was the first American war in which more soldiers died of enemy action than disease. Prior to that, wars were races to see who could get the most men to battle field before everyone dropped dead from dysentery, plague or whatever.
If fending off an invasion on his own land in what was likely to be a couple of brief battles, nobles could sometimes scrape up 15% or more of the farmers but largely because the farmers wanted to fend off the plunder of the invading army.
The end of the knights and aristocrats began when the growing urban population began to deploy large armies of well trained infantry units in the later 1300s (see the battle of the spurs.) These urban areas could sometime raise 50%-75% of the military age males, already equipped and trained and alliance of a couple of cities or more could easily out raise even the kings of the time. Their weakness lay in their inability to fight protracted wars at a distance without wrecking the economy.
The Swiss solved this problem by hiring out some men when they were young as mercenaries. It made money, got the boys of troublesome age out of town till they settled down, left a battle harden cadre of elders behind and in the worst case, they could call the mercenaries back. Ruled European battlefields for nearly 150 years with pikes and crossbows.
Arguably, every major empire in history arose when a very small society gained a significant advantage in internal cohesion and instead of worrying about internal revolution, could arm their entire adult male population if that is what it took. That is how the small backwoods town of Rome became an Empire. Conversely, when the Republic feel and the Legions chose the Emperors, trust broke down and an Empire with something like 30-40 times the population of the Republic couldn't deploy armies a tenth as large as the Republic did repeatedly in the Punic wars.
So, if you want to calculate the size of army a medieval society could raise, the primary factor is how the farmer majority and the urbanites view the enemy. If the commoners see the war as just more dynastic struggles, then the armies will be small, usually in the thousands, mostly aristocrats and mercenaries.
If they faced something like an invasion of Mongols, Ottomans or Vikings, who the commoners feared for their own sakes, then the armies could be quite large, tens of thousands, assuming the fighting occurred locally in summer.
Peasant rebellions would also raise up surprising amounts of man power if conditions were bad enough. See the Hussite rebellions.
(It's worth noting that the Mongols and Vikings attack with very small forces, relying on speed and maneuver to overwhelm local opposition before the levies could be called up. The so-called Mongol "Hordes" were usually outnumbered 10 to 1, but they rode so fast they seemed to be everywhere. Same thing for vikings. They could hit several places on the coast before the central land power heard of the first creating the impression of multiple forces hitting simutanously.)
Maximum: ~7,000 soldiers
Presuming a high household size of 2.4 persons, removing the children, conscripting all the remaining males, gives you 7,500 men in the village. Removing 500 old fuddy-duddys, leaves you with 7,000 able-bodied males (presuming women weren't allowed to fight). Women and children take over the remaining work to be done.
Speculative Size: ~50 soldiers
During peacetime, with no looming threats more than a bandit here and there, I would use your standing force as a police force. In relatively low-crime, modern cities of the same size, such as Woodburn, Oregon (chosen as a similar size, and relatively isolated); you have 35 patrol officers. However, they are assisted by some state troopers and the city can always be assisted the National Guard, military, etc. if the need arrives.
In your city, your 50 soldiers are acting like a police patrol force as well as a standing military in peacetime.
During medieval England, there was a requirement for every man and boy to practice archery - should a war with France requrire additional soldiers.
https://en.wikipedia.org/wiki/Unlawful_Games_Act_1541
The law stated that all boys from 7 to 17 would be provided a bow and arrows by the parents (to practice) and from 17 the boy would be required to provide his own (to fight).
Training was required every week; and playing other games was forbidden.
Assuming an even distribution of ages in the village, aged 0 - 60; and an age of 20-60 able to fight; this leaves 2/3 of the male population that can be called upon - leading to 2/3 * 1/2 * 25,000 ~ 8,300 men.
Note that these will be just the archers that can be called upon, and more expensive soldiers with armour or horses will be significantly less. This also ignores any sort of morale impact - as calling every male from a village is likely to have a negative impact on the rulers ability to control said village.
The Ancient, not Medieval Roman Army supported 300000 in an empire of 30 million plus. Medieval states, being less organized, could not match this proportion.
What makes a Medieval state a Medieval state is that there is more or less zero organizational ability at the high level. There is no standing army per se, rather, the state is broken into little districts and each soldier is supported by the farmers in that district. He rules that area so that he can get the supplies, and 'owes' the leadership support in war. Bandit suppression is done by these local forces, if they care to. In real history, often the efficiency in getting everyone to show up for a war was extremely limited.
Then the other factor comes in - if you call in all the nobles and they show up, how do you feed them? Medieval states had no real logistic support. They had to rent boats to sail across the waters, for example. Troops were literally expected to show up with some weeks food, and when that ran out, they went home.
A city of 25000 is pretty large for medieval times, and you might have a set of professional knights and squires of 50-100 permanently available. If the city was attacked they could round up regular joes and put them on the walls for defense of a sort. You might have a squad of archers about. Remember, the side coming to attack your city probably only has a few thousand men in it themselves.
Remember, the Merovingian Empire (France and Germany) and Anglo Saxon England were both basically helpless for a few generations from Viking raiders.
There are a lot of factors.
Food. Medieval era had subsistence farming. It was once an odd concept to not own your own farm. How would you make sure you have enough to eat?
If the lord could centralize farming, have specialist farmers (again a ludicrous concept where someone does nothing but farming), you can get very high food productivity.
Training. How much time does it take to train your average soldier? The Romans were very systematic in training. Yet some things, like English longbowmen took years to train. A knight who was trained to fight since puberty by the best mentors will be able to defeat many militias who were trained in a few weeks.
How good do they need to be? Ex-military bandits can easily take on regular militia. Even regular bandits who are motivated by self-wealth can beat many inexperienced militia.
Equipment. Who will link the chainmail together? Who will make the weapons? Melt the steel bars? Mine the ore? Transport it?
A 'military outpost' where much of this work is outsourced can be extremely military oriented. The larger the town is, the less people need to actually do this work, but there will be other overhead to a large city, like sewage.
So, it can easily be anything from 1% to 90%. It depends on how many of these factors play in.
Consider that war is a seasonal pursuit. As others have mentioned a full time standing army was not really a medieval thing.
The planting and harvest seasons require quite a lot of manpower, so would not be a time to go to war.
Summer has more available manpower, unless local crops requires regular bucket-line irrigation. But heat may restrict how far you can march your forces.
Winter will tend to have the most free people available, but weather and temperature may prevent it.
An "optimal" time to go to war would be immediately after fields have been harvested and storehouses have been filled and before winter storms begin to set in. This gives you some benefits.
