In several places in the Bible, the sky is referred to as a vault, with the stars stuck on it. Genesis 1 refers to water above this vault. (The Babylonian cosmology also placed water there.) The Book of Revelation states that the stars will someday fall out of the sky like figs from a tree. The Bible says little about the shape of the Earth, referring in one place to the "circle" of the Earth, and in another place to the "four corners" of the Earth. In one of the Gospels, the Devil tempted Jesus by taking him up a mountain where he could see "all the kingdoms of the world". The mountain is not named, but it is sometimes argued that this requires a flat earth. Similarly, in Daniel, there is a vision of a tree "visible to the earth's furthest bounds".
The Bible does indicate more clearly, however, that the Earth is motionless. Witness Joshua's telling the Sun (and not the Earth) to stop just so he could win one of his battles, and some of the Psalms that state that the Earth is motionless.
In all fairness, this cosmology was perfectly reasonable when the Old Testament was written. It wasn't until about 300 BC that the Greeks determined the shape of the earth, and measured its size. (The Greeks still had the Sun move around the Earth, though.)
There are fundamentalists who find all this to be compelling. I haven't found any of them the Web (sorry) but they do exist. They are aware of being scoffed at, but their particular 'literal' interpretation of the Bible is important to them.
The odd thing is that some Scientific Creationists don't repudiate these people. I mean, there are getting to be a lot of photos taken from outer space. It's spherical, guys.