'web owner' may be in need of disambiguation. I assume this is a web hosting company.
If so, they have a duty to provide some security, but, a) their terms and conditions very probably and properly exclude liability. Check.
b) No security in such situations can be guaranteed. See the current attacks on the major 'web owner' by toasters.
Nearly last, but by no means least - see Atti above.
My own contribution is that you must never leave responsibility for your data with a third party. It will go wrong often enough for you to need your own backups.