You post batches of things to them (you can forward email receipts too). They scan and do some data entry: the name of the biller and the total owing and some other things so you can search later. Cheapest plan seems to be $20 a month so sadly not very cheap. It's aimed more at businesses but we were really losing control of our filing so we went with it. There's a free plan but only if you upload scans yourself, they won't process the postal batches.

It takes about a week for a scanned batch to be online. After that it just invokes logging into their website, searching, and downloading the PDF you need. I guess 30 seconds?