It's definitely conceivable.
Light, as we see it, is in the wavelength band of 390 to 700 nm. This is a rather narrow band of light, but it's the only band you need to reflect to make your world dark. All the rest can come in as normal to heat the planet. And you wouldn't even need to completely reflect all the visible light. You wanted constant dusk, so you only need to stop 70-90%(ish) of the light from reaching Earth. The rest could be reflected... Or absorbed. (For more about that, see articles describing global warming.) When volcanoes erupt, they spew a bunch of reflective stuff into the atmosphere. In big eruptions, this usually lowers the global temperature for a year or so.
Great, there's some background. Now how do I do it?
The first thing you'd wanna do is set up a big black cloud in the lower atmosphere, preferably of a greenhouse gas. This will absorb light from the sun and heat up. 184
Unfortunately, as <PERSON> pointed out in comments, this absorbs a lot of heat, so much that your planet will cook. Well, how do you fix that? Mirrors. Right above the black cloud, if you set up another cloud, but this time use something like sulfur to block the light This will allow only a set percentage of the light to hit your black fog.
So... You've successfully destroyed our beautiful sky. What's that gonna do to the planet? (I swear, I plead NPOV.) For one, your plants will change. If they don't die outright, they'll turn black. Since visible light is in such short supply, they'll try to capture as much of it as possible. Let's assume that your plants survived without a significant hit to the $O_2$ in the atmosphere. Great, what's next? Wind. As your cloud stops most of the light from reaching Earth, but traps heat, the ground won't radiate heat like it does today. This will alter or stop the cyclical motion of the lowest atmosphere, causing weird effects.
In summary
While it is definitely possible to selectively alter the climate (indeed, we can do it today!) be careful of unintended effects that result from it. 184