Early 2000's anime has definitely aged. In my opinion the aesthetic of it has aged extremely well, but I understand that the relatively poor animation and picture quality are enough to turn some people off to it. If you like the aesthetic as much as I do, then I have a whole list. I'm not much of a shonen guy, so a lot of these are calmer, less action-packed shows.
My personal favorite anime is Haibane Renmei, from 2002. The story seems like a mystery at first, but the show never answers most of the questions it raises–it's not about finding answers, it's about the characters, and how they act and react both towards each other and towards the world they live in. The setting is what really pulled me in–it has the same timeless, liminal feel that Ico, Shadow of the Colossus, and The Last Guardian have.
Cowboy Bebop and Trigun are staples. Both 1998 shows. It's hard to believe that Cowboy Bebop was made in the 20th century. Both shows have good plotlines, lovable characters, well-made action sequences. It's hard to write anything about them that hasn't been written before. Watch em!
Witch Hunter Robin from 2002 was pretty good, imo. It's a mystery series about a team of witch hunters. It starts out as a sort of monster (or witch, as it were) of the week show before pivoting into a longer story arc about the characters discovering a conspiracy. The show looks drab and gloomy at first, but really it's absolutely dripping with character, and Amon is the only person in the show that never really seems happy. The action sequences are largely made up of characters just sort of standing and staring at each other, but the camera work and visual effects make it look really good in my opinion.
Fullmetal Alchemist '03 was the first FMA I watched when I was a kid, but never got around to finishing until last year. It's really good. Brotherhood and '03 are largely similar (with minor differences) up until the 5th Laboratory, where they diverge severely. Personally I felt like '03 had a more concise, well-told story. Having fewer characters worked in its favor, because it gave the characters that it does have more time to develop. It also takes itself a lot more seriously than Brotherhood–it feels more like a seinen than a shonen.
Planetes (also from 2003) is the hardest sci-fi that I think I've ever seen. It's about team of orbital debris haulers in the 2070s. At first it's a slice of life show about their day to day activities, and partway through it becomes something of a political drama/action show. It's cool. The main character is a little bit annoyingly idealistic in the first few episodes, but after that it's a solid 10/10 from me.
Last Exile (also 2003) is one of my favorite shows. It's very confusing at first, it feels like there's a couple of episodes worth of exposition missing in the middle, but all in all I absolutely adored it. It never gives you any more information than you need to know, and it never wastes time explaining how its universe works. How do vanships fly? Because of claudia. What is claudia? It's the fuel that makes vanships fly. You never even find out what Exile means until like 23 episodes in. The show is extremely aware of its own aesthetic, in a way that the sequel series from 2011 kind of wasn't. Still worth watching both series IMO, because the sequel finds its own footing in time.
Mushishi (2005) is another calm one. Probably the calmest one. Watching it is like meditating. It follows a sort of travelling doctor who tries to help people when their interactions with the ethereal mushi (they are explicitly living things, but for the sake of storytelling, you can think of them as spirits) turn harmful. Every episode is a new place, a new mushi, a new story. It's masterfully crafted, and the slow pace ensures that not a single frame goes to waste. The early 2000's aesthetic works heavily in its favor, making it fit right in with Studio Ghibli's works, even though it was an Artland joint.
Baccano was 2007, so a bit late to call it "turn of the century," but it has the same sort of aesthetic and vibe that most of these shows have. The story has a wide variety of characters, each with their own fully fleshed out stories, all intertwining like a spider's web as they meet and influence each other. It's really good. Watch the dub!
Right now I'm nearing the end of Noir (2002). I couldn't tell you if it's worth watching until I finish it (just in case it ends badly), but so far I'm liking it. It's about a pair of assassins, one of whom has amnesia and only knows that she's somehow connected to the other, and the other who is trying to find out who killed her family and why. If I have one gripe with the series, it's that they don't ever show any blood on screen. They aren't afraid of showing people die on screen, a lot of people die in every episode, but aside from holes appearing in people's clothing, there doesn't appear to be any actual violence.