"DNA data" comes in several forms dependent on the technology used to produce them.
Companies like 23andMe are using SNP chips and those are available for an only rather a limited number of species. To be completely honest, I don't even know what would be the raw data they would provide, but I would suspect it could be a vcf file.
If the chosen technology would be DNA sequencing. DNA sequencing is provided by a few dozen of sequencing centres around the world, some of them associated with universities. DNA sequencing generates files that contain individual sequencing reads with assigned quality scores ("how much the sequencer trusts each base"), the typical formats are
fastq (Illumina) or
.bam (long read technologies). Usually, you would also need to find a suitable reference in one of the public databases, for instance, the NCBI portal for human reference. Then map your reads and do variant calling. Of course, then it depends on what would you be interested in.
If you don't care for your own DNA and you are more interested in life around you, you can check ncbi for all sorts of genomic data. If you would be interested to understand how the raw sequencing data work, ebi is a good place to look. Also, in every single paper generating/using sequencing data you there should be mention where in public databases you can find them.
Finally, let me warn you. Some of the genomic analyses are quite computationally heavy (not all though). If you don't have a specific question in mind, it would be wise to start with bacterial sequencing.
Hope you will have fun with DNA data!