How to disable possibility to enter past date in activities?
I think you'll need an extension for this.  Use the hook_civicrm_validateForm if you just need to prevent prior dates from being entered on the back-end activity form.  If you need to block ALL forms of adding a prior date, you may want to use the hook_civicrm_pre hook (this will block Webform-CiviCRM, API, etc.).

You may want to set exclusions for administrators if you do this!