This question is a little problematic to answer because what survives today is not necessarily the same as what was common 600+ years ago. What was "common" was unlikely to have been preserved because it was used as a working tool. We wouldn't know conclusively exactly how polished an average weapon would have been because so few of the "average weapons" survive in mint condition, not just because of use, but because of time. An average weapon likely would have been used until it was unusable, sold and the metal re-purposed for something else. Else it was lost, and it has rusted away.

This is also subject to the Caveat that "medieval" is a huge time period itself with many different areas that had different resources, different skills etc. And you vastly increase the scope by bringing "greek/roman" times.

So let's limit the question in scope and talk about the late medieval period in England and western Europe. Roughly the 15th century, although a fair number of the examples below stretch into the 16th century.

And the answer to your question is, absolutely, yes. Late medieval blacksmiths could produce forged weapons that are of a quality comparable to what you'd see on TV or movies. The catch is that it took them much longer, and consequently, the pieces tended to be much more expensive, reflecting the hours of work that went into them. This means, of course, that cheaper weapons would naturally have been less polished or less ground.

Surprisingly, they did have many of the things we have today, but with different means of locomotion. Blacksmiths had power hammers driven by water wheels, and we know, for example, that the Freibergdorf Hammer Mill was founded around 1607. Older mills existed but they do not survive.

Likewise, replacing a modern belt grinder, they had pedal driven and handle driven grinders and apprentices to use them. The Luttrell Psalter dating to about 1430 describes in some detail a grindstone with two hand cranks on each end of the axle to be used for driving. This is a much later example of a waterwheel driven grindstone

It's also important to note that like many other things, swords were tools. A "working" sword in the possession of a warrior would be well maintained but might not be kept "pretty" and would accumulate dents and nicks if used in combat. On the other hand, ceremonial weapons would have been prettier, more detailed, AND are more likely to have been preserved to the present because they were as much works of art as they were ever meant to be working weapons. However, I don't think it matters for the purposes of your question. A high quality ceremonial weapon shows the skills of blacksmiths of the era at their height. If they chose to produce less quality work for lower cost, that was a decision they could make and their customers can make. Spears and axes and mauls require less metal and less work than swords, and were far more common as mass infantry weapons.

So let's look at some examples of well preserved relic weapons. Like I said, many of these are ceremonial weapons.

This article contains pictures of "blessed swords" presented by various popes to catholic rulers as gifts. On page 39 is a photograph of a sword presented to the Archduke Ferdinand in 1582 and some drawings of swords presented in the 15th century.

The Manx sword of state is a 15th century design probably made in London, but experts think it was fitted with a new blade in the 16th or 17th century. Given that it's ceremonial but not "royal" it may be more indicative of common quality in that era.

Here are color photographs of the swedish swords of state which was commissioned in 1541, and the second around 1560.

This is the Reichsswert, the imperial sword of the holy roman empire, kept in the Imperial Treasury in Vienna dating to the early 14th century at least and possibly earlier.

Although not nearly well preseved, there are approximately 170 so called "ulfberth swords" that carry a common trade mark. approximately 44 have been found in Norway, and 31 in Finland, but many of the examples are fragmentary. The earliest example is thought to date to the 9th or 10th century and later examples date to the 12th century. This is a close up picture of one found in Kazan in the Volga river valley that is a later model. although it has been rusted and the hilt had rotted away, you can get an idea of what it likely looked like in mint condition.

This is likewise a picture in a museum of other viking era swords from the 10th-11th century. they have also been rusted, but you can get an idea of teh craftsmanship. Some hammer marks are evident, but they likey would have been quite polished when new.

There are really two questions here - could medieval craftsmen make armour and weapons that was 'perfect' to our modern eyes, and the other question is - did they? In short, they could, but they mostly didn't. Just because medieval craftmen had the means to create symmetrical pieces doesn't mean that they thought that was important, out of all the practical and aesthetic concerns they had to deal with.

I know you asked about swords, but I'm going to talk quite a bit about armour here - partly because it is my own specialty (weapons are a bit of a side gig) and partly because a lot more has been written about the armouring industry than the swordmaking industry. However, some broad outlines apply to both. Moreover, I'm going to focus on the later middle ages and early modern period, which are better documented but also produce a lot of iconic artifacts that we think of as 'medieval' - the industries of the early middle ages were quite different!

It is actually pretty simple to figure see what armourers and swordsmiths were capable of - we can look at the surviving artifacts that they produced. For instance, here is a 15th century English Hanger probably for an infantryman like an archer. Note the simplicity of the guard, the slight asymmetries of the blade, and the general lack of 'polish' - even though it is an archaeological find, you can still tell that the original wasn't necessarily a weapon that had been lovingly finished. But it is a weapon that has been finished as much as it needs to be in order to do its job, and no more. You see somewhat better quality but the same 'is it good enough'? approach in many of the swords from the largest late medieval sword-find, the so called 'Castillon swords' found in the River Dordogne, which may have been English swords intended for or used by the army at Castillon at 1453 (they sunk in a barge and were found 500 years later). For example this and this - both swords are better finished than the hanger, and may have been carried by infantry or by men at arms (knights or those who fought as knights). Even so there are slight imperfections and assymetries that don't just result from them being in the water for 500 years. They were better decorated and finished than the hanger, but not necessarily made symmetrical.

In armour the assymetries and irregularities of premodern workmanship are much more obvious. Indeed, precise workmanship is pretty unusual in late medieval and early modern plate armour, enough that it really stands out when you see it - it looks weird once you're used to the usual wonkiness. My favorite example of just how freaking weird looking late medieval workmanship can get are infantry breastplates from the later 15th century. Like this one* or this one or my personal favorite, this one (seriously, what the fuck, guys?). So if you want to picture an average professional soldier (a retainer of a lord or a mercenary) you can perhaps picture him wielding a functional and well balanced hanger that looks a little off and a breastplate with assymetrical fluting and a cockeyed plackart.

However, asymmetries don't stop with the munition armour of common soldiers - even very expensive armours made for clients with a lot of florins have all sorts of oddities - vision slits that are wider at one end than the other, flutes that don't quite line up, that kind of thing. Nothing that takes away from the function of the armour, and when decorated nothing that actually takes away from its aesthetics, but still not the 'perfect' look that we are used to seeking. Even in the 'high end' armours of the 15th century, the symmetry of a few masters like Lorenz Helmschmied is almost unnerving.

So...why are premodern European weapons and armour so...imprecise, if not crude? You mentioned how many power tools that modern smiths have, and no doubt that helps make precise finishes much easier. Finishing armour, historically, was labor intensive and expensive - up to 80% of the labor that went into a man at arm's full harness would go into polishing alone. Power hammers make plannishing every single minor divet and dent out of armour much easier, certainly, and powered polishing wheels help too. But it should be said that late medieval armourers probably had access to both water-powered polishing wheels (in the later 15th century the Duke of Milan gifted a polishing mill to the Missaglia family of armourers, to whom he owed hundreds of thousands of florins) and also water-powered trip hammers. It is clear that polishing mills were used on finished armour, while it is likely but less clear to me that trip-hammers were used in making finished armours (as opposed to flattening plates so that they could be worked by armourers). Still, neither of these tools would be as precise or easy to use as modern power tools, of course. On the other hand, while modern smiths are working alone (and indeed, that is part of the romance of modern artisanal crafts in general), late medieval swordsmiths and armourers worked in large workshops that divided labor between different tasks, so that polishers focused on polishing, locksmiths dealt with buckles, etc. Alternately the manufacture of individual pieces and the assembly of the finished product coudl be separated between different workshops, so you might have bladesmiths make a blade, ship it overseas, and local cutlers put it in its hilt according to the local fashion. But while later medieval workshops had a lot of labor, they also had to produce a -lot- of goods, often on tight schedules (if there is a war coming, you need to crank out product pretty fast). So the incentive was very much to make something that was 'good enough.'

But finally, I don't think that late medieval people necessarily had the same eye toward perfect precision that we do - our eyes have been trained by machined exactness, and that effects our impression of what 'good quality' craftsmanship is - to us, the best craftsman is an exact one, which is to say the one who is able to do by hand what machines can do. On the other hand, Ruskin in the 19th century identified 'crudeness' (meaning a lack of symmetry) as a hallmark of medieval craft. For him, it was the result of craftsmen working at the very limits of their talents to glorify God, as opposed to the industrially produced decorative knicknacks of his own time. While that may be a scathing critique of the state of labor in industrial capitalism, it isn't a very accurate description of actual medieval workmanship, which was driven more by the prosaic concerns of economics, time, and the eyes of their consumers.

*(I have handled this one in person, it is even cruder than the pictures show but it is very solid - it feels like it would get the job done)

One very good response already.

This comment by one of the moderators (u/WARitter) discussing production and logistics of medieval weaponry might also interest you.

it got me thinking, in every piece of entertainment set in medieval/greek/roman times I can recall people run around with flawless swords and spears.

This is partly because most of the weapons get very little close-up screen time. Sometimes, a special weapon gets significant screen time, such as Excalibur being the star of some shots in Boorman's Excalibur (1981), but even in the movie, those shots are fairly brief and can have action that distracts from close examination of the sword (e.g., the scene where Uryens knights Arthur in the moat, mid-battle).

Battle scenes tend to be chaotic, and many weapons are seen in close-up, but usually only very briefly, and with plenty of action to distract you from looking for flaws in the weapons.

Secondly, typical "flaws" in weapons will often be hard to see. Things like off-centre ridgelines on gladius blades, asymmetric fullers, guards at slight angles to the blades, wavy blade surfaces from either poor polishing or forging marks not removed are not very visible from a distance.

Spear hafts are one thing that could show "flaws", like lack of straightness (a flaw, sometimes) or non-uniformity. But it like most modern movie spear hafts use turned dowel, and tend be quite straight (historical hafts were typically heat-straightened if necessary, so tended to be quite straight, too). Some of the spear hafts seen in Troy (2004) appear to be not-quite-straight.

u/AncientHistory has already mentioned that prop weapons might be simply and cheaply made of sheet aluminium or steel, to produce something that looks enough like a sword from a distance. Casting in resin (often fibreglass reinforced) is also common.

In Conan the Barbarian (1982), two swords featured in multiple close-ups: Conan's two swords. 4 of each were made, and were used in close-ups. The ones mainly used in fight scenes were resin and aluminium. Similar things are done in other movies: where weapons will feature in close-ups, the better-looking ones are used. This can increase the difficulty of doing scenes, since the better-looking ones are often (unsharpened) steel, but where it will matter, and if the budget allows, one sees the better swords. Likewise, the least-authentic looking props are often in the background of battle scenes, where many soldiers are on screen, and little of the weapons can be seen - it is enough for the prop to look vaguely like a sword or spear.

Some weapons with deliberate flaws are sometimes used. This is often to show some group as peasants, barbarians, orcs, etc.

u/BigBennP noted that

And the answer to your question is, absolutely, yes. Late medieval blacksmiths could produce forged weapons that are of a quality comparable to what you'd see on TV or movies. The catch is that it took them much longer, and consequently, the pieces tended to be much more expensive, reflecting the hours of work that went into them. This means, of course, that cheaper weapons would naturally have been less polished or less ground.

and u/WARitter made a similar point:

There are really two questions here - could medieval craftsmen make armour and weapons that was 'perfect' to our modern eyes, and the other question is - did they? In short, they could, but they mostly didn't. Just because medieval craftmen had the means to create symmetrical pieces doesn't mean that they thought that was important, out of all the practical and aesthetic concerns they had to deal with.

u/WARitter also noted that good enough is good enough:

It is actually pretty simple to figure see what armourers and swordsmiths were capable of - we can look at the surviving artifacts that they produced. For instance, here is a 15th century English Hanger probably for an infantryman like an archer. Note the simplicity of the guard, the slight asymmetries of the blade, and the general lack of 'polish' - even though it is an archaeological find, you can still tell that the original wasn't necessarily a weapon that had been lovingly finished. But it is a weapon that has been finished as much as it needs to be in order to do its job, and no more.

Where swords are made as artworks, the appearance matters a lot. This can include gilding, bluing, engraving, etc. It can include a high level of symmetry, but doesn't always, because the weapon needs fabulous from a distance, not under close examination. Where swords are made as weapons, function matters. Minor asymmetries in blades, gaps between blade and guard (both of which draw complaints from modern buyers of swords) matter very little. If the blade is of suitable weight, balance, and geometry, and the edge is sharp enough, and the steel good enough, it is functional. (Of course, it can be intended as both artwork and functional weapon.)

Old swords vary a lot in weight and balance, even among a particular type of sword (e.g., there are Viking Age swords of 600-700g, and at least three at the rather monstrous weight of 1.8-1.9kg). A Viking Age sword differs from a 19th century cavalry sabre which differs from a smallsword which differs from a Roman gladius. There are plenty of solutions to the "good enough" design, and the diversity of the historical solutions strongly suggests that there is no perfect design of sword.

If we assume that there is no perfect sword, we should keep in mind that it is quite possible to make inferior swords, design-wise. A big part of this is the handling of the sword. We don't know the details of how all swords from the past were used, so it's difficult to be sure that some particular sword is guilty of bad handling. But we know that some types of swords appear to be (usually) designed to give good handling, balanced to give the desired point control (more a matter of the location of the pivot point AKA centre of percussion than the point of balance). Weight is often a compromise between metallurgy, specifically the strength of steel, and reach and lightness. Modern replicas, especially if long and heavy, are sometimes very poorly-handling because they put too much weight near the tip, Better-designed historical swords - which can be a joy to wield compared to such clunky replicas - show that sword-makers of the past knew what they were doing design-wise. In terms of design, they could achieve function on a par with the best modern makers. Given that the handling of old swords, even of the same type varies a lot, it's likely that at least part of this variation is a variation in quality (and some is due to personal preference).

It is in terms of metallurgy that a modern makers has huge potential advantage. Working with modern steels of known composition (and techniques such as mass spectroscopy and XRF to check their composition) and known processes for good heat treatment, and the tools to do that heat treatment can let a modern maker produce swords of superb metallurgical quality.

A major function of traditional sword-making with laminated blades is to make pre-modern heat treatment more reliable (especially tempering, reducing the brittleness of the blade after quenching). Also, differential hardening. Both lamination and differential hardening are famously used in traditional Japanese sword-making. Rather than being special Japanese techniques, they were widely used around the world, and in some cases still are. (Most Medieval European differential hardening of swords appears to have been by slack-quenching, using the residual heat to auto-temper the blade.)

At the level of metallurgy, there was a great deal in variation in quality, and many weapons were far short of what was possibly. Iron swords (with unhardened blades) were used alongside pattern-welded swords with sometimes quite hard steel edges and alongside crucible steel swords which were often left unhardened (because ultra-high carbon steels can get very brittle indeed). Clearly, all met a reasonable standard of "good enough", for some customer.

So "good enough" is not just about finish and ornamentation. It goes to the core of the design of swords - does the sword work well enough? It also goes to the core of the sword itself - is the steel and its heat treatment good enough?

Some further reading:

Peter Johnsson and Vincent Le Chevalier, "Documenting the dynamics of swords", 2017: http://blog.subcaelo.net/ensis/documenting-dynamics-of-swords/

On the diversity of Medieval European swords, and how late all-steel construction replace lamination: Williams, Alan, Sword and the Crucible: A History of the Metallurgy of European Swords up to the 16th Century, BRILL, 2012.