I began to wonder about instruments in the time period of the popular classical composers - were the instruments as refined as they are now?

That depends on what you mean by "refined". They wouldn't have had modern materials and manufacturing techniques available to them, of course, but within that context instrument-makers and performers were just as skilled and dedicated to their craft as they are now. There was a great culture of virtuoso instrumental performance as far back as the 17th century, as well. Obviously, the lack of recording technology prior to the very late 19th century means we can't make any real side-by-side comparisons of ineffable things like technical proficiency and expressiveness, but it's not unreasonable to assume that people 200-300 years ago had the same kinds of aesthetic sensibilities as we do even if the specific details of taste and style have changed. We can look at the music itself, too: the Bach sonatas and partitas for solo violin (before 1720), Vivaldi's countless concertos (mostly before 1730), Paganini's caprices (1802-1817), and Liszt's Transcendental Etudes (1826) remain as technically challenging for modern performers as they were when they were written.

Orchestral stringed instruments have been pretty much modern in their physical form since the 16th century — a non-specialist would be hard-pressed to distinguish this 1658 Stainer violin from a newly-made instrument with a mere visual inspection — but metal strings are an early 20th century invention. Prior to that, orchestral musicians would have been playing on gut strings. Chinrests were invented in the early 19th century by violinist/composer Louis Spohr (~15 years younger than Beethoven). As I'm sure you're aware, 17th-century Italian-made stringed instruments — especially Stradivari and Guarneri, but also Amati and Bergonzi to a lesser degree — are generally considered superior to anything crafted before or since.

Wind and brass instruments have changed a lot in the last ~300 years. For instance, a Baroque flute looks and sounds very different from a modern orchestral flute. Valves for brass instruments weren't invented until the early 19th century, so the instruments Beethoven knew look like this and have a somewhat grittier sound than a modern horn. I won't get into the physics of brass instruments unless you really want to know; suffice to say that natural brass like the horn linked above only have access to a limited set of pitches through changes in embouchure (the shape and tension of the player's lips). Players could change the pitches they had access to by adding or removing sections of tubing called crooks. Modern valved brass, by contrast, are fully chromatic except in their lowest registers (trombones are a bit of a special case because of the slide).

There are many, many musicians and orchestras that specialize in informed period performance. Mass media examples (like Immortal Beloved or Amadeus) are almost always going to use performances with modern instruments, though, because it's cheaper and the general public isn't going to notice the difference. As far as "nicely made" or "sound[ing] as good", that's often a matter of opinion, but there's certainly no shortage of beautifully-crafted period instruments of all kinds. Period performances definitely have some different tone colors, but a good performance is a good performance: Brandenburg Concertos.