is there a way a document can be saved on the blockchain?

inscribe it as an ordinal

https://bitcoinfiles.org/ has been around for a long time

for example https://bitcoinfiles.org/t/f9e6c4f0ac7219257e1276cd23c1bff5e5088204ff4e3471786c6252fb00f01e

does a private file only remain seen only by the uploader?