To have something at the L-1 Lagrange point (directly between the sun and planet) that's large enough to cast a shadow, it would have to be very large and of negligible mass to be even reasonably stable. It wouldn't leave an area in perpetual darkness, though; as the planet rotated under it, the shadow would move, albeit in a predictable path.

However, if your goal is to have a body that leaves a particular area in darkness of similar size to a lunar eclipse here on Earth (100ish km diameter full dark and 6400ish km partial darkness), then the interposing body would have to possess quite an unusual orbital path (that is, it would effectively be a powered vessel of some sort). The area in darkness would basically just be what you'd get with perpetual night: not much in the way of plant life and fairly cold relative to the surrounding areas. Such as spot would probably wreak havoc with local weather and wildlife. However, there might be a thriving tourist industry of folks coming to dispose of items there ("Stick it where the sun don't shine" would be a pilgrimage of sorts rather than merely a rude suggestion). It would be a lot like the traditional underdark in the center, but with open skies.

In the Ringworld books, there are very large floating cities tethered in place (on the order of miles across, if I recall). The areas under these cities were in perpetual darkness, with similar effects to the above. A major difference was the presence of sewers and other things emptying down from the city above onto the landscape, leading to a whole culture of dark-adapted night people who disposed of the dead and other such things.