Many hosts that do make backups, only make them for internal use, so a restore will cost the customer some money. There's virtually no host out there that guarantees the availability of data/backup, so, the only real safe way is to make your own backups as often as necessary.
They say they simply "don't have backups"
At least they don't beat around the bush.

In any case, they're a budget provider, some corners have to be cut, and apparently, backups were one.
Sorry to hear you put your trust on them being available. I can only imagine the nightmare that losing data can be.
As per the rules, can you share a domain that you are/have been hosting with them? You can just send it to me by PM.