A universe having a finite volume can be unbounded in length and have unbounded cross-sectional area.
The example I have in mind is mathematical, not physical. It's called Gabriel's Horn. It's a standard example in first-year calculus.
It's also called Toricelli's Trumpet, after Evangelista Torricelli, a student of Galileo. His discovery of this strange mathematical object set off a philosophical storm about infinity back in the 1640's. He is most famous for inventing the barometer.
You generate Gabriel's Horn by rotating the graph of y = 1/x about the x-axis, and considering the horn-shaped region swept out for values of x >= 1. (You steer way clear of the messy bit at zero where it blows up). Using basic calculus you can show that it has infinite surface area and infinite cross-sectional area but finite volume.
In fact, its volume is ... (drum roll, please) ... pi.
If you lived in a universe shaped like this, your greatest physicists would determine that the volume of your universe is finite. Yet the universe would contain a 2-dimensional cross-section whose area is infinite; and the boundary of the universe would have infinite surface area. But the volume is finite, and that's the best definition of a finite universe.
As I visualize this, there's a 2-D boundary of infinite surface area, bounding a finite volume. That's what Gabriel's Horn is.
Now, in this universe, which remember has finite volume, you can jump down the hole down the middle (along the x-axis) and keep on going forever. You can increase your x-coordinate without bound. So yes this is a very skinny hole! It's one-dimensional. I don't know if that will satisfy the OP's idea of "bottomless pit." Surely a tiny little point-creature could slither down a line. Maybe one of Deleuze's infinitesimals, who's to say?
So that would be my concept of a finite universe with a bottomless pit.
(ps) You know, the pit isn't one-dimensional at all. At every finite value of x the cross-section has positive area. So a little particle or being could vibrate down the hole, just at a smaller and smaller wavelength ... hence higher and higher energy ... so a physicist would have to jump in and invoke quantumness to spoil all the fun. But an object doesn't have to be infinitely skinny to get down this hole ... it just has to get skinnier and skinnier, the farther down it goes. But it's always got nonzero size, just tending to zero at the limit.
And in fact at every point inside the horn, you could fit a little three-dimensional being. So maybe this isn't so bad at all. In order to keep falling down the hole you just have to keep getting smaller. But you can continue to be a three-dimensional creature all the way down ... just one that's getting smaller.
I think this example works. It's a finite universe with an infinitely deep pit.