Motivation: FASTQ is a standard file format for DNA sequencing data which stores both nucleotides and quality scores. A typical sequencing study can easily generate hundreds of gigabytes of FASTQ files, while public archives such as ENA and NCBI and large international collaborations such as the Cancer Genome Atlas can accumulate many terabytes of data in this format. Text compression tools such as gzip are often employed to reduce the storage burden, but have the disadvantage that the data must be decompressed before it can be used. Here we present BEETL-fastq, a tool that not only compresses FASTQ-formatted DNA reads more compactly than gzip, but also permits rapid search for $k$-mer queries within the archived sequences. Importantly, the full FASTQ record of each matching read or read pair is returned, allowing the search results to be piped directly to any of the many standard tools that accept FASTQ data as input. Results: We show that 6.6 terabytes of human reads in FASTQ format can be transformed into 1.7 terabytes of indexed files, from where we can search for 1, 10, 100, 1000, a million of 30-mers in respectively 3, 8, 14, 45 and 567 seconds plus 20 ms per output read. Useful applications of the search capability are highlighted, including the genotyping of structural variant breakpoints and "in silico pull-down" experiments in which only the reads that cover a region of interest are selectively extracted for the purposes of variant calling or visualization. Availability: BEETL-fastq is part of the BEETL library, available as a github repository at email@example.com:BEETL/BEETL.git.