Yes, with the qualifier that it only looks like a spandex suit.
The suit would have an outer layer that is basically reinforced pseudo-spandex. Weave some sort damage resistant fiber around highly elastic fiber and surface everything with brightly colored and reflective dye.
The elastic fiber makes it act like spandex, the other fiber gives it damage resistance and the brightly colored dye makes the surface look smoother than it is by hiding shadows caused by unevenness. This is also convenient since it is an excuse to have suits with bright colors. The suits might even come with small LED system to boost the effect.
Under the spandex looking layer would be the actual armor. Structurally it would be plate mail composed of interlocking plates of varying size. Plates would be connected with actual hinges that would let them move fairly freely within the limits needed for normal movement. Small springs would maintain tight fit without compromising flexibility.
With the tight fit and interlocking plates the outer layer should be sufficient to hide the relatively small gaps between the plates. And as long as movements stay within what the armor is designed to handle there would be no visible extra rigidity or looseness. And it is not really not an issue that the suit is more rigid than your body if somebody tries to break your limbs or neck.
There are several alternatives for the materials. Steel or reinforced plastic gives adequate protection for your needs and machining steel or 3D printing the plastic can be done by computer controlled machines which simplifies making the custom fitted plates this type of armor requires.
I'd probably use plates made of two steel plates with rubber in between in manner similar to non-explosive reactive armor. This wouldn't really give resistance to shaped charges or anything but the movement of the outer steel layer would absorb some of the energy without breaking anything and it should be possible to make the joints between plates lock when the outer plate moves spreading the impact to neighbouring plates.
Under this would want some sort of well breathing padding to make wearing the suit bearable and further absorb any impacts.
This should give you a bullet proof armor that looks like spandex suit. It would also make you look really bulky and probably weigh a lot. It wouldn't really impact your movement since the extra weigh would be well balanced and tightly fitted to your body but you probably would not look like somebody who'd be expected to want to show off their body with a spandex suit. It might get embarrassing if people get the idea that you are some sort of a role model because you defy the "thin is beautiful" ideology.
I am only mentioning the looks issue since it is kind of important to the genre. But the amount of thickness obviously corresponds to the amount of armour protection and the plates can be shaped to exaggerate your musculature so it might be fine?