Biological meaning of read length
The read length has absolutely nothing to do with what you are sequencing. It is a characteristic of the sequencing technology you use. NGS sequencing techniques typically produce this sort of short read you're seeing. The read length does not change because you are sequencing a longer molecule. You would still get ~250nt reads even if you were sequencing an entire genome. Your reads are something like this (image source):



So the vast majority of your 250nt are overlapping and cover slightly different parts of your target sequence. This is one of the reasons why NGS analysis is not trivial. The first step in any NGS analysis is to assemble your reads into a bam file that covers your target region. If you need help on doing that, come on over to http://bioinformatics.stackexchange.com.